Graph Quadratic Functions by Completing the Square

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Question 1 of 4

1. Question

Graph the function

$y=x^2-10x+15$

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Vertex Form

$y=a(x-h)^2+k$

where

$(h,k)$ is the vertex

Completing the square is done by producing a square of a binomial on the left side of the equal sign. This method is useful when no two rational numbers solve the equation.

Plot the vertex and

$x$ intercepts, then connect the points to form a parabola

First, find the vertex and plot this on the graph

Start by transforming the function into vertex form

$y$	$=$	$x^2-10x+15$
$y$ $-15$	$=$	$x^2-10x+15$ $-15$	Subtract $15$ from both sides
$y-15$	$=$	$x^2-10x$

Take the coefficient of the

$x$ term, divide it by two and then square it.

$y-15$	$=$	$x^2$ $-10$ $x$			Coefficient of the $x$ term

	$=$	$\frac{\color{#00880A}{-10}}{2}$			Divide it by $2$

		$(-5)^2$	$=$	$25$	Square

This number will make the right side a perfect square.

Add and subtract $25$ to the right side to keep the balance.

$y-15$	$=$	$x^2-10x$
$y-15$	$=$	$x^2-10x+$ $25$ $-$ $25$	Add and subtract $25$

Now, transform the right side into a square of a binomial, then leave

$y$ on the left side.

$y-15$	$=$	$(x-5)(x-5)-25$
$y-15$	$=$	$(x-5)^2-25$
$y-15$ $+15$	$=$	$(x-5)^2-25$ $+15$
$y$	$=$	$(x-5)^2-10$

The function is now in vertex form

Compare the function to the general vertex form to get the vertex

$y$	$=$	$a(x-h)^2+k$

$y$	$=$	$(x-5)^2-10$

$h$

$=$

$5$

$k$

$=$

$-10$

This means that the vertex is at

$(5,-10)$

Next, find the

$x$ intercepts by substituting

$y=0$ , then solving for

$x$

$y$	$=$	$(x-5)^2-10$
$0$	$=$	$(x-5)^2-10$	Substitute $y=0$
$0$ $+10$	$=$	$(x-5)^2-10$ $+10$	Add $10$ to both sides
$10$	$=$	$(x-5)^2$
$sqrt10$	$=$	$sqrt((x-5)^2)$	Take the square root of both sides
$+-sqrt10$	$=$	$x-5$
$x-5$	$=$	$+-sqrt10$
$x-5$ $+5$	$=$	$+-sqrt10$ $+5$	Add $5$ to both sides
$x$	$=$	$5+-sqrt10$
$x$	$=$	$8.162, 1.838$

Mark these

$2$ points on the

$x$ axis

Finally, connect the points to form a parabola

Question 2 of 4

2. Question

Graph the function

$y=-2x^2-8x+2$

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Vertex Form

$y=a(x-h)^2+k$

where

$(h,k)$ is the vertex

Completing the square is done by producing a square of a binomial on the left side of the equal sign. This method is useful when no two rational numbers solve the equation.

Plot the vertex and

$x$ intercepts, then connect the points to form a parabola

First, find the vertex and plot this on the graph

Start by transforming the function into vertex form

$y$	$=$	$-2x^2-8x+2$
$y$	$=$	$-2(x^2+4x-1)$	Factor out $-2$

Take the coefficient of the

$x$ term, divide it by two and then square it.

$y$	$=$	$-2(x^2+$ $4$ $x-1)$			Coefficient of the $x$ term
	$=$	$\frac{\color{#00880A}{4}}{2}$			Divide it by $2$

		$(2)^2$	$=$	$4$	Square

This number will make the right side a perfect square.

Add and subtract $4$ to the grouping of

$x$ to keep the balance.

$y$	$=$	$-2(x^2+4x-1)$
$y$	$=$	$-2(x^2+4x+$ $4$ $-$ $4$ $-1)$	Add and subtract $4$
$y$	$=$	$-2((x+2)^2-4-1)$
$y$	$=$	$-2((x+2)^2-5)$
$y$	$=$	$-2(x+2)^2+10$

This is now in vertex form

Compare the function to the general vertex form to get the vertex

$y$	$=$	$a(x-h)^2+k$
$y$	$=$	$-2(x+2)^2+10$

$h$

$=$

$-2$

$k$

$=$

$10$

This means that the vertex is at

$(-2,10)$

Next, find the

$x$ intercepts by substituting

$y=0$ , then solving for

$x$

$y$	$=$	$-2(x+2)^2+10$
$0$	$=$	$-2(x+2)^2+10$	Substitute $y=0$
$0$ $-10$	$=$	$-2(x+2)^2+10$ $-10$	Subtract $10$ from both sides
$-10$	$=$	$-2(x+2)^2$
$-10$ $-:(-2)$	$=$	$-2(x+2)^2$ $-:(-2)$	Divide both sides by $-2$
$5$	$=$	$(x+2)^2$
$(x+2)^2$	$=$	$5$
$sqrt((x+2)^2)$	$=$	$sqrt5$	Take the square root of both sides
$x+2$	$=$	$+-sqrt5$
$x+2$ $-2$	$=$	$+-sqrt5$ $-2$	Subtract $2$ from both sides
$x$	$=$	$-2+-sqrt5$

Mark these

$2$ points on the

$x$ axis

Finally, connect the points to form a parabola

Question 3 of 4

3. Question

By completing the square, which graph is correct for the equation:

$y=x^2-8x+24$ .

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Completing the square is done by producing a square of a binomial on the left side of the equal sign. This method is useful when no two rational numbers solve the equation.

Perform the process of completing the square on the given quadratic to convert into vertex form.

$y$	$=$	$x^2-8x+24$
	$=$	$(x^2-8x$ $+ (-8/2)^2$ $)$ $-(-8/2)^2$ $+ 24$	Complete the square

	$=$	$(x^2-8x$ $+16$ $)$ $-16$ $+ 24$	Simplify

	$=$	$(x^2-8x$ $+16$ $)+8$
$y$	$=$	$(x-4)^2+8$	Rewrite as a square of a binomial

Identify the vertex of the graph from the given formula.

$y$	$=$	$a(x-\color{blue}{h})^2+\color{blue}{k}$
$y$	$=$	$(x-4)^2+8$	Given equation
$y$	$=$	$a(x-\color{blue}{4})^2+\color{blue}{8}$	Extract values of $h$ and $k$
Vertex	$=$	$(\color{blue}{h},\color{blue}{k})$
Vertex	$=$	$(\color{blue}{4},\color{blue}{8})$

Mark the vertex on the graph.

Next, solve for the

$y$ -intercept by substituting $x=0$ .

$y$	$=$	$(x-4)^2+8$
$y$	$=$	$(0-4)^2+8$	Substitute $x=0$
$y$	$=$	$16+8$
$y$	$=$	$24$

Mark the

$y$ -intercept on the graph.

Draw a parabola using the points.

Question 4 of 4

4. Question

By completing the square, which graph is correct for the equation:

$y=x^2-8x+18$ .

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Completing the square is done by producing a square of a binomial on the left side of the equal sign. This method is useful when no two rational numbers solve the equation.

Perform the process of completing the square on the given quadratic to convert into vertex form.

$y$	$=$	$x^2-8x+18$
	$=$	$(x^2-8x$ $+ (-8/2)^2$ $)$ $-(-8/2)^2$ $+ 18$	Complete the square

	$=$	$(x^2-8x$ $+16$ $)$ $-16$ $+ 18$	Simplify

	$=$	$(x^2-8x$ $+16$ $)+2$
$y$	$=$	$(x-4)^2+2$	Rewrite as a square of a binomial

Identify the vertex of the graph from the given formula.

$y$	$=$	$a(x-\color{blue}{h})^2+\color{blue}{k}$
$y$	$=$	$(x-4)^2+2$	Given equation
$y$	$=$	$a(x-\color{blue}{4})^2+\color{blue}{2}$	Extract values of $h$ and $k$
Vertex	$=$	$(\color{blue}{h},\color{blue}{k})$
Vertex	$=$	$(\color{blue}{4},\color{blue}{2})$

Mark the vertex on the graph.

Next, solve for the

$y$ -intercept by substituting $x=0$ .

$y$	$=$	$(x-4)^2+2$
$y$	$=$	$(0-4)^2+2$	Substitute $x=0$
$y$	$=$	$16+2$
$y$	$=$	$18$

Mark the

$y$ -intercept on the graph.

Draw a parabola using the points.

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Powers of a Power 3
Powers of a Power 4
Zero Powers 1
Zero Powers 2
Negative Exponents 1
Negative Exponents 2
Negative Exponents 3
Rational Exponents 1
Rational Exponents 2
Rational Exponents 3
Mixed Operations with Exponents 1
Mixed Operations with Exponents 2
Exponent Notation 1
Exponent Notation 2
Exponent Notation 3
Multiply Exponents (Product Rule) 1
Multiply Exponents (Product Rule) 2
Multiply Exponents (Product Rule) 3
Multiply Exponents (Product Rule) 4
Divide Exponents (Quotient Rule) 1
Divide Exponents (Quotient Rule) 2
Powers of a Power 1
Powers of a Power 2
Powers of a Power 3
Powers of a Power 4
Zero Powers 1
Zero Powers 2
Negative Exponents 1
Negative Exponents 2
Negative Exponents 3
Rational Exponents 1
Rational Exponents 2
Rational Exponents 3
Mixed Operations with Exponents 1
Mixed Operations with Exponents 2
Exponent Notation 1
Exponent Notation 2
Exponent Notation 3
Multiply Exponents (Product Rule) 1
Multiply Exponents (Product Rule) 2
Multiply Exponents (Product Rule) 3
Multiply Exponents (Product Rule) 4
Divide Exponents (Quotient Rule) 1
Divide Exponents (Quotient Rule) 2
Powers of a Power 1
Powers of a Power 2
Powers of a Power 3
Powers of a Power 4
Zero Powers 1
Zero Powers 2
Negative Exponents 1
Negative Exponents 2
Negative Exponents 3
Find the Hypotenuse 1
Find the Hypotenuse 2
Find the Hypotenuse 3
Find a Side 1
Find a Side 2
Find a Side 3
Pythagoras Problems 1
Pythagoras Problems 2
Pythagoras Problems 3
Pythagoras Mixed Review 1
Pythagoras Mixed Review 2
Pythagoras Mixed Review 3
Pythagoras Mixed Review 4
Find the Hypotenuse 1
Find the Hypotenuse 2
Find the Hypotenuse 3
Find a Side 1
Find a Side 2
Find a Side 3
Pythagoras Problems 1
Pythagoras Problems 2
Pythagoras Problems 3
Pythagoras Mixed Review 1
Pythagoras Mixed Review 2
Pythagoras Mixed Review 3
Pythagoras Mixed Review 4
Complementary and Supplementary Angles 1
Complementary and Supplementary Angles 2
Complementary and Supplementary Angles 3
Vertical, Revolution and Reflex Angles 1
Vertical, Revolution and Reflex Angles 2
Alternate, Corresponding and Co-Interior Angles 1
Alternate, Corresponding and Co-Interior Angles 2
Alternate, Corresponding and Co-Interior Angles 3
Angles and Parallel Lines
Triangle Geometry 1
Triangle Geometry 2
Triangle Geometry 3
Quadrilateral Geometry 1
Quadrilateral Geometry 2
Ratios 1
Ratios 2
Ratios 3
Ratios 4
Proportions 1
Proportions 2
Dividing Quantities
Rates 1
Rates 2
Rates 3
Rates 4
Scales 1
Scales 2
Scales 3
Vertical Translations (Shifts) 1
Vertical Translations (Shifts) 2
Vertical Translations (Shifts) from a Point
Horizontal Translations (Shifts) 1
Horizontal Translations (Shifts) from a Point
Horizontal Translations (Shifts) from a Graph
Horizontal and Verticals Translations (Shifts) from a Graph
Sketch a Graph using Translations (Shifts)
Write the Equation from a Graph
Write the Equation from Translations (Shifts) 1
Vertical Dilations (Stretch/Shrink)
Horizontal Dilations (Stretch/Shrink) 1
Horizontal Dilations (Stretch/Shrink) 2
Horizontal Dilations (Stretch/Shrink) – Scale Factor
Horizontal and Vertical Dilations (Stretch/Shrink) 1
Horizontal and Vertical Dilations (Stretch/Shrink) 2
Horizontal and Vertical Dilations (Stretch/Shrink) 3
Graphing Reflections 1
Graphing Reflections 2
Reflection with Rotation
Combinations of Transformations: Order
Combinations of Transformations: Coordinates
Combinations of Transformations: Find Equation 1
Combinations of Transformations: Find Equation 2
Combinations of Transformations: Find Equation 3
Simplify Square Roots 1
Simplify Square Roots 2
Simplify Square Roots 3
Simplify Square Roots 4
Simplify Radicals with Variables 1
Simplify Radicals with Variables 2
Simplify Radicals with Variables 3
Rewriting Entire and Mixed Radicals 1
Rewriting Entire and Mixed Radicals 2
Add and Subtract Radical Expressions (Basic) 1
Add and Subtract Radical Expressions (Basic) 2
Add and Subtract Radical Expressions (Basic) 3
Add and Subtract Radical Expressions 1
Add and Subtract Radical Expressions 2
Add and Subtract Radical Expressions 3
Multiply Radical Expressions 1
Multiply Radical Expressions 2
Multiply Radical Expressions 3
Multiply Radical Expressions 4
Divide Radical Expressions 1
Divide Radical Expressions 2
Divide Radical Expressions 3
Multiply and Divide Radical Expressions
Simplify Radical Expressions using the Distributive Property 1
Simplify Radical Expressions using the Distributive Property 2
Simplify Radical Expressions using the Distributive Property 3
Simplify Binomial Radical Expressions using the FOIL Method 1
Simplify Binomial Radical Expressions using the FOIL Method 2
Rationalizing the Denominator 1
Rationalizing the Denominator 2
Rationalizing the Denominator 3
Rationalizing the Denominator 4
Rationalizing the Denominator using Conjugates
Simplify Square Roots 1
Simplify Square Roots 2
Simplify Square Roots 3
Simplify Square Roots 4
Simplify Radicals with Variables 1
Simplify Radicals with Variables 2
Simplify Radicals with Variables 3
Rewriting Entire and Mixed Radicals 1
Rewriting Entire and Mixed Radicals 2
Add and Subtract Radical Expressions (Basic) 1
Add and Subtract Radical Expressions (Basic) 2
Add and Subtract Radical Expressions (Basic) 3
Add and Subtract Radical Expressions 1
Add and Subtract Radical Expressions 2
Add and Subtract Radical Expressions 3
Multiply Radical Expressions 1
Multiply Radical Expressions 2
Multiply Radical Expressions 3
Multiply Radical Expressions 4
Divide Radical Expressions 1
Divide Radical Expressions 2
Divide Radical Expressions 3
Multiply and Divide Radical Expressions
Simplify Radical Expressions using the Distributive Property 1
Simplify Radical Expressions using the Distributive Property 2
Simplify Radical Expressions using the Distributive Property 3
Simplify Binomial Radical Expressions using the FOIL Method 1
Simplify Binomial Radical Expressions using the FOIL Method 2
Rationalizing the Denominator 1
Rationalizing the Denominator 2
Rationalizing the Denominator 3
Rationalizing the Denominator 4
Rationalizing the Denominator using Conjugates
Add & Subtract Matrices 1
Add & Subtract Matrices 2
Add & Subtract Matrices 3
Multiply Matrices 1
Multiply Matrices 2
Matrices: Multiplication Word Problems
Determinant of a Matrix
Inverse of a Matrix
Matrices: Systems of Equations 1
Matrices: Systems of Equations 2
Gauss Jordan Elimination
Cramer’s Rule
Solve a System of Equations by Graphing
Substitution Method 1
Substitution Method 2
Substitution Method 3
Substitution Method 4
Elimination Method 1
Elimination Method 2
Elimination Method 3
Elimination Method 4
Systems of Nonlinear Equations
Systems of Equations Word Problems 1
Systems of Equations Word Problems 2
3 Variable Systems of Equations – Substitution Method
3 Variable Systems of Equations – Elimination Method
Solve a System of Equations by Graphing
Substitution Method 1
Substitution Method 2
Substitution Method 3
Substitution Method 4
Elimination Method 1
Elimination Method 2
Elimination Method 3
Elimination Method 4
Systems of Nonlinear Equations
Systems of Equations Word Problems 1
Systems of Equations Word Problems 2
3 Variable Systems of Equations – Substitution Method
3 Variable Systems of Equations – Elimination Method
Greatest Common Factor 1
Greatest Common Factor 2
Factor Expressions using GCF
Factor Expressions 1
Factor Expressions 2
Factor Expressions with Negative Numbers
Factor Difference of Two Squares 1
Factor Difference of Two Squares 2
Factor Difference of Two Squares 3
Factor by Grouping
Factor Difference of Two Squares (Harder) 1
Factor Difference of Two Squares (Harder) 2
Factor Difference of Two Squares (Harder) 3
Factor Quadratics 1
Factor Quadratics 2
Factor Quadratics 3
Factor Quadratics with Leading Coefficient more than 1 (1)
Factor Quadratics with Leading Coefficient more than 1 (2)
Factor Quadratics with Leading Coefficient more than 1 (3)
Factor Quadratics (Complex)
Solve Quadratics by Factoring 1
Solve Quadratics by Factoring 2
The Quadratic Formula
Completing the Square 1
Completing the Square 2
Intro to Quadratic Functions (Parabolas) 1
Intro to Quadratic Functions (Parabolas) 2
Intro to Quadratic Functions (Parabolas) 3
Graph Quadratic Functions in Standard Form 1
Graph Quadratic Functions in Standard Form 2
Graph Quadratic Functions by Completing the Square
Graph Quadratic Functions in Vertex Form
Write a Quadratic Equation from the Graph
Write a Quadratic Equation Given the Vertex and Another Point
Quadratic Inequalities 1
Quadratic Inequalities 2
Combining Methods for Solving Quadratic Equations
Convert Between Logarithmic and Exponent Form 1
Convert Between Logarithmic and Exponent Form 2
Evaluate Logarithms 1
Evaluate Logarithms 2
Evaluate Logarithms 3
Expand Log Expressions
Simplify Log Expressions 1
Simplify Log Expressions 2
Simplify Log Expressions 3
Logarithmic Equations 1
Logarithmic Equations 2
Logarithmic Equations 3
Change Of Base Formula
Solving Exponential Equations Using Log Laws
Simplify Roots of Negative Numbers 1
Simplify Roots of Negative Numbers 2
Powers of the Imaginary Unit 1
Powers of the Imaginary Unit 2
Solve Quadratic Equations with Complex Solutions 1
Solve Quadratic Equations with Complex Solutions 2
Equality of Complex Numbers
Add and Subtract Complex Numbers 1
Add and Subtract Complex Numbers 2
Multiply Complex Numbers 1
Multiply Complex Numbers 2
Divide Complex Numbers
Complex Numbers – Product of Linear Factors 1
Complex Numbers – Product of Linear Factors 2
Mixed Operations with Complex Numbers
One Step Inequalities 1
One Step Inequalities 2
Two Step Inequalities
Multi-Step Inequalities 1
Multi-Step Inequalities 2
Compound Inequalities 1
Compound Inequalities 2
Compound Inequalities 3
Inequality Word Problems 1
Inequality Word Problems 2
One Step Inequalities 1
One Step Inequalities 2
Two Step Inequalities
Multi-Step Inequalities 1
Multi-Step Inequalities 2
Compound Inequalities 1
Compound Inequalities 2
Compound Inequalities 3
Inequality Word Problems 1
Inequality Word Problems 2
Absolute Value Inequalities
Graph Linear Inequalities 1
Graph Linear Inequalities 2
Simplify Algebraic Expressions 1
Simplify Algebraic Expressions 2
Simplify Algebraic Expressions 3
Substitution and Evaluating Algebraic Expressions
Substitution and Evaluating Multi-Variable Algebraic Expressions
Substitution and Evaluating Rational Algebraic Expressions
Substitution and Evaluating Algebraic Expressions with Negative Numbers 1
Substitution and Evaluating Algebraic Expressions with Negative Numbers 2
Simplify Algebraic Expressions using the Distributive Property 1
Simplify Algebraic Expressions using the Distributive Property 2
Simplify Algebraic Expressions using the Distributive Property 3
Write Algebraic Expressions from Diagrams
Evaluate Algebraic Expressions Problems
Combine Like Terms 1
Combine Like Terms 2
Combine Like Terms 3
Simplify Algebraic Expressions 1
Simplify Algebraic Expressions 2
Simplify Algebraic Expressions 3
Substitution and Evaluating Algebraic Expressions
Substitution and Evaluating Multi-Variable Algebraic Expressions
Substitution and Evaluating Rational Algebraic Expressions
Substitution and Evaluating Algebraic Expressions with Negative Numbers 1
Substitution and Evaluating Algebraic Expressions with Negative Numbers 2
Simplify Algebraic Expressions using the Distributive Property 1
Simplify Algebraic Expressions using the Distributive Property 2
Simplify Algebraic Expressions using the Distributive Property 3
Write Algebraic Expressions from Diagrams
Evaluate Algebraic Expressions Problems
Combine Like Terms 1
Combine Like Terms 2
Combine Like Terms 3
Compass Bearings and True Bearings 1
Compass Bearings and True Bearings 2
Solving for Bearings
Bearings from Opposite Direction
Using Bearings to Find Distance 1
Using Bearings to Find Distance 2
Using Bearings to Find Distance 3
Using Bearings and Distances to Find Angles
Working with Radial Surveys 1
Working with Radial Surveys 2
Working with Radial Surveys 3
Working with Radial Surveys 4
Multiply Binomials
Multiply Binomials – FOIL Method
FOIL Method – Same First Variable 1
FOIL Method – Same First Variable 2
FOIL Method – 3 Terms
Expand Perfect Squares
Difference of Two Squares
Difference of Two Squares (Longer Expressions)
Greatest Common Factor 1
Greatest Common Factor 2
Factor Expressions using GCF
Factor Expressions 1
Factor Expressions 2
Factor Expressions with Negative Numbers
Factor Difference of Two Squares 1
Factor Difference of Two Squares 2
Factor Difference of Two Squares 3
Factor by Grouping
Factor Difference of Two Squares (Harder) 1
Factor Difference of Two Squares (Harder) 2
Factor Difference of Two Squares (Harder) 3
Factor Quadratics 1
Factor Quadratics 2
Factor Quadratics 3
Factor Quadratics with Leading Coefficient more than 1 (1)
Factor Quadratics with Leading Coefficient more than 1 (2)
Factor Quadratics with Leading Coefficient more than 1 (3)
Factor Quadratics (Complex)
Convert Measurements (Length) 1
Convert Measurements (Length) 2
Convert Measurements (Volume) 1
Convert Measurements (Volume) 2
Convert Measurements (Mass) 1
Convert Measurements (Mass) 2
Add & Subtract Matrices 1
Add & Subtract Matrices 2
Add & Subtract Matrices 3
Multiply Matrices 1
Multiply Matrices 2
Matrices: Multiplication Word Problems
Determinant of a Matrix
Inverse of a Matrix
Matrices: Systems of Equations 1
Matrices: Systems of Equations 2
Gauss Jordan Elimination
Cramer’s Rule
Add and Subtract Negative Numbers on the Number Line 1
Add and Subtract Negative Numbers on the Number Line 2
Add and Subtract Negative Numbers on the Number Line 3
Add and Subtract Negative Numbers on the Number Line 4
Add and Subtract Negative Numbers 1
Add and Subtract Negative Numbers 2
Add and Subtract Negative Numbers 3
Add and Subtract Negative Numbers 4
Multiply and Divide Negative Numbers 1
Multiply and Divide Negative Numbers 2
Multiply and Divide Negative Numbers 3
Negative Numbers: Order of Operations
Negative Numbers with Variables 1
Negative Numbers with Variables 2
Negative Numbers with Variables 3
Negative Numbers with Substitution 1
Negative Numbers with Substitution 2
Negative Numbers with Substitution 3
Negative Numbers Word Problems
Convert To and From Scientific Notation
Multiply & Divide in Scientific Notation
Scientific Notation Word Problems
Find the Hypotenuse 1
Find the Hypotenuse 2
Find the Hypotenuse 3
Find a Side 1
Find a Side 2
Find a Side 3
Pythagoras Problems 1
Pythagoras Problems 2
Pythagoras Problems 3
Pythagoras Mixed Review 1
Pythagoras Mixed Review 2
Pythagoras Mixed Review 3
Pythagoras Mixed Review 4
Increase and Decrease an Amount by a Percent – Word Problems 1
Increase and Decrease an Amount by a Percent – Word Problems 2
Percent of Change
Percent of Change – Word Problems
Percent of an Amount – Word Problems 1
Percent of an Amount – Word Problems 2
Ratios 1
Ratios 2
Ratios 3
Ratios 4
Proportions 1
Proportions 2
Dividing Quantities
Rates 1
Rates 2
Rates 3
Rates 4
Scales 1
Scales 2
Scales 3
Increase and Decrease an Amount by a Percent – Word Problems 1
Increase and Decrease an Amount by a Percent – Word Problems 2
Percent of Change
Percent of Change – Word Problems
Percent of an Amount – Word Problems 1
Percent of an Amount – Word Problems 2
Find Mean, Mode, Median and Range 1
Find Mean, Mode, Median and Range 2
Find Mean, Mode, Median and Range 3
Create Frequency Tables & Graphs
Interpret Frequency Tables 1
Interpret Frequency Tables 2
Create and Interpret Bar & Line Graphs (Histograms)
Interpret Cumulative Frequency Tables and Charts 1
Interpret Cumulative Frequency Tables and Charts 2
Create Grouped Frequency Tables and Graphs
Interpret Grouped Frequency Tables
Create and Interpret Dot Plots (Line Plots) 1
Create and Interpret Dot Plots (Line Plots) 2
Finding the Interquartile Range 1
Finding the Interquartile Range 2
Create and Interpret Box & Whisker Plots 1
Create and Interpret Box & Whisker Plots 2
Create and Interpret Box & Whisker Plots 3
Create and Interpret Box & Whisker Plots 4
Create and Interpret Stem & Leaf Plots 1
Create and Interpret Stem & Leaf Plots 2
Create and Interpret Stem & Leaf Plots 3
Create and Interpret Stem & Leaf Plots 4
Mean and Standard Deviation
The Normal Curve 1
The Normal Curve 2
The Normal Curve 3
Z-Values (Z-Scores)
Vertices and Edges
Degrees 1
Degrees 2
Degrees 3
Drawing a Network 1
Drawing a Network 2
Completing a Table from a Network Diagram
Network from Maps and Plans
Identify Paths and Cycles
Eulerian Trails and Circuits 1
Eulerian Trails and Circuits 2
Identify Spanning Trees
Minimum Spanning Trees 1
Minimum Spanning Trees 2
Shortest Path 1
Shortest Path 2
Convert Units of Time 1
Convert Units of Time 2
Convert Units of Time 3
Add Units of Time
Subtract Units of Time
Convert 12 Hour Time to 24 Hour Time 1
Convert 12 Hour Time to 24 Hour Time 2
Convert 12 Hour Time to 24 Hour Time 3
Convert 24 Hour Time to 12 Hour Time 1
Convert 24 Hour Time to 12 Hour Time 2
Add and Subtract Time Using the Calculator
Time Difference and Longitudes 1
Time Difference and Longitudes 2
Time Difference and Longitudes 3
International Time Zones 1
International Time Zones 2
Time and Travelling Word Problems 1
Time and Travelling Word Problems 2
Great Circle Distances
FOIL Method – 3 Terms
Division of Polynomials
Remainder Theorem
Factor Theorem
Graphing Polynomials
Graph Polynomials
Solve a System of Equations by Graphing
Substitution Method
Elimination Method
Systems of Nonlinear Equations
Systems of Equations Word Problems
3 Variable Systems of Equations – Substitution Method
3 Variable Systems of Equations – Elimination Method
Convert Radians and Degrees 1
Convert Radians and Degrees 2
Convert Radians and Degrees 3
Radians and Arc Length (Radians)
Areas of Sectors and Segments (Radians)
The Unit Circle
Unit Circle: Find Angles
Unit Circle: Trig Ratios
Unit Circle: Exact Values 1
Unit Circle: Exact Values 2
Unit Circle: Find Angles from Equations
Intro to Trigonometric Ratios (SOH CAH TOA) 1
Intro to Trigonometric Ratios (SOH CAH TOA) 2
Round Angles (Degrees, Minutes, Seconds)
Evaluate Trig Expressions using a Calculator 1
Evaluate Trig Expressions using a Calculator 2
Trig Ratios: Solving for a Side 1
Trig Ratios: Solving for a Side 2
Trig Ratios: Solving for an Angle
Angles of Elevation and Depression
Trig Ratios Word Problems: Solving for a Side
Trig Ratios Word Problems: Solving for an Angle
Area of Non-Right Angled Triangles 1
Area of Non-Right Angled Triangles 2
Law of Sines: Solving for a Side
Law of Sines: Solving for an Angle
Law of Cosines: Solving for a Side
Law of Cosines: Solving for an Angle
Trigonometry Word Problems 1
Trigonometry Word Problems 2
Trigonometry Mixed Review: Part 1 (1)
Trigonometry Mixed Review: Part 1 (2)
Trigonometry Mixed Review: Part 1 (3)
Trigonometry Mixed Review: Part 1 (4)
Trigonometry Mixed Review: Part 2 (1)
Trigonometry Mixed Review: Part 2 (2)
Trigonometry Mixed Review: Part 2 (3)
Intro to Trigonometric Ratios (SOH CAH TOA) 1
Intro to Trigonometric Ratios (SOH CAH TOA) 2
Round Angles (Degrees, Minutes, Seconds)
Evaluate Trig Expressions using a Calculator 1
Evaluate Trig Expressions using a Calculator 2
Trig Ratios: Solving for a Side 1
Trig Ratios: Solving for a Side 2
Trig Ratios: Solving for an Angle
Angles of Elevation and Depression
Trig Ratios Word Problems: Solving for a Side
Trig Ratios Word Problems: Solving for an Angle
Area of Non-Right Angled Triangles 1
Area of Non-Right Angled Triangles 2
Law of Sines: Solving for a Side
Law of Sines: Solving for an Angle
Law of Cosines: Solving for a Side
Law of Cosines: Solving for an Angle
Trigonometry Word Problems 1
Trigonometry Word Problems 2
Trigonometry Mixed Review: Part 1 (1)
Trigonometry Mixed Review: Part 1 (2)
Trigonometry Mixed Review: Part 1 (3)
Trigonometry Mixed Review: Part 1 (4)
Trigonometry Mixed Review: Part 2 (1)
Trigonometry Mixed Review: Part 2 (2)
Trigonometry Mixed Review: Part 2 (3)
Intro to Trigonometric Ratios (SOH CAH TOA) 1
Intro to Trigonometric Ratios (SOH CAH TOA) 2
Round Angles (Degrees, Minutes, Seconds)
Evaluate Trig Expressions using a Calculator 1
Evaluate Trig Expressions using a Calculator 2
Trig Ratios: Solving for a Side 1
Trig Ratios: Solving for a Side 2
Trig Ratios: Solving for an Angle
Angles of Elevation and Depression
Trig Ratios Word Problems: Solving for a Side
Trig Ratios Word Problems: Solving for an Angle
Area of Non-Right Angled Triangles 1
Area of Non-Right Angled Triangles 2
Law of Sines: Solving for a Side
Law of Sines: Solving for an Angle
Law of Cosines: Solving for a Side
Law of Cosines: Solving for an Angle
Trigonometry Word Problems 1
Trigonometry Word Problems 2
Trigonometry Mixed Review: Part 1 (1)
Trigonometry Mixed Review: Part 1 (2)
Trigonometry Mixed Review: Part 1 (3)
Trigonometry Mixed Review: Part 1 (4)
Trigonometry Mixed Review: Part 2 (1)
Trigonometry Mixed Review: Part 2 (2)
Trigonometry Mixed Review: Part 2 (3)
Convert Radians and Degrees 1
Convert Radians and Degrees 2
Convert Radians and Degrees 3
Radians and Arc Length (Radians)
Areas of Sectors and Segments (Radians)
The Unit Circle
Unit Circle: Find Angles
Unit Circle: Trig Ratios
Unit Circle: Exact Values 1
Unit Circle: Exact Values 2
Unit Circle: Find Angles from Equations
Find Base from Percent of an Amount (Unitary Method) 1
Find Base from Percent of an Amount (Unitary Method) 2
Using Percentages with Proportions
Find Original Amount Before Percent Change (Unitary Method)
Find Base from Percent of an Amount (Unitary Method) 1
Find Base from Percent of an Amount (Unitary Method) 2
Using Percentages with Proportions
Find Original Amount Before Percent Change (Unitary Method)
Derivative of Trigonometric Functions 1
Derivative of Trigonometric Functions 2
Derivative of Trigonometric Functions 3
Trig Applications of Differentiation
Integral of Trigonometric Functions 1
Integral of Trigonometric Functions 2
Trig Applications of Integration
Graph Trigonometric Functions 1
Graph Trigonometric Functions 2
Graph Trigonometric Functions 3
Graph Trigonometric Functions 4
Derivative of Trigonometric Functions 1
Derivative of Trigonometric Functions 2
Derivative of Trigonometric Functions 3
Trig Applications of Differentiation
Integral of Trigonometric Functions 1
Integral of Trigonometric Functions 1
Trig Applications of Integration
Graph Trigonometric Functions 1
Graph Trigonometric Functions 2
Graph Trigonometric Functions 3
Graph Trigonometric Functions 4
Power Rule 1
Power Rule 2
Power Rule 3
Power Rule 4
Chain Rule 1
Chain Rule 2
Product Rule
Quotient Rule
Derivatives of Exponential Functions 1
Derivatives of Exponential Functions 2
Derivatives of Exponential Functions 3
Derivatives of Trigonometric Functions 1
Derivatives of Trigonometric Functions 2
Derivatives of Trigonometric Functions 3
Power Rule 1
Power Rule 2
Power Rule 3
Power Rule 4
Chain Rule 1
Chain Rule 2
Product Rule
Quotient Rule
Derivatives of Exponential Functions 1
Derivatives of Exponential Functions 2
Derivatives of Exponential Functions 3
Derivatives of Trigonometric Functions 1
Derivatives of Trigonometric Functions 2
Derivatives of Trigonometric Functions 3
Derivatives of Exponential Functions 1
Derivatives of Exponential Functions 2
Derivatives of Exponential Functions 3
Integrals of Exponential Functions
Definite Integrals of Exponential Functions
Definite Integrals of Logarithmic Functions
Indefinite Integrals of Logarithmic Functions 1
Indefinite Integrals of Logarithmic Functions 2
Direct Variation
Inverse Variation 1
Inverse Variation 2
Indefinite Integrals 1
Indefinite Integrals 2
Indefinite Integrals 3
Indefinite Integrals of Exponential Functions
Indefinite Integrals of Logarithmic Functions 1
Indefinite Integrals of Logarithmic Functions 2
Indefinite Integrals of Trig Functions
Definite Integrals
Definite Integrals of Exponential Functions
Definite Integrals of Logarithmic Functions
Definite Integrals of Trig Functions
Areas Between Curves and the Axis 1
Areas Between Curves and the Axis 2
Area Between Curves
Volumes of Revolution 1
Volumes of Revolution 2
Volumes of Revolution 3
Trapezoidal Rule
Simpsons Rule
Applications of Integration for Trig Functions
One Step Inequalities 1
One Step Inequalities 2
Two Step Inequalities
Multi-Step Inequalities 1
Multi-Step Inequalities 2
Compound Inequalities 1
Compound Inequalities 2
Compound Inequalities 3
Inequality Word Problems 1
Inequality Word Problems 2
Absolute Value Inequalities
Graph Linear Inequalities 1
Graph Linear Inequalities 2
Multiply Binomials
Multiply Binomials – FOIL Method
FOIL Method – Same First Variable 1
FOIL Method – Same First Variable 2
FOIL Method – 3 Terms
Expand Perfect Squares
Difference of Two Squares
Difference of Two Squares (Longer Expressions)
Multiply Binomials
Multiply Binomials – FOIL Method
FOIL Method – Same First Variable 1
FOIL Method – Same First Variable 2
FOIL Method – 3 Terms
Expand Perfect Squares
Difference of Two Squares
Difference of Two Squares (Longer Expressions)
Multiply Binomials
Multiply Binomials – FOIL Method
FOIL Method – Same First Variable 1
FOIL Method – Same First Variable 2
FOIL Method – 3 Terms
Expand Perfect Squares
Difference of Two Squares
Difference of Two Squares (Longer Expressions)
Solving Exponential Equations
Factorial Notation
Fundamental Counting Principle 1
Fundamental Counting Principle 2
Fundamental Counting Principle 3
Combinations 1
Combinations 2
Combinations with Restrictions 1
Combinations with Restrictions 2
Combinations with Probability
Basic Permutations 1
Basic Permutations 2
Basic Permutations 3
Permutation Problems 1
Permutation Problems 2
Permutations with Repetitions 1
Permutations with Repetitions 2
Permutations with Restrictions 1
Permutations with Restrictions 2
Permutations with Restrictions 3
Permutations with Restrictions 4
Distance Between Two Points 1
Distance Between Two Points 2
Distance Between Two Points 3
Midpoint of a Line 1
Midpoint of a Line 2
Midpoint of a Line 3
Slope of a Line 1
Slope of a Line 2
Slope Intercept Form: Graph an Equation 1
Slope Intercept Form: Graph an Equation 2
Slope Intercept Form: Write an Equation 1
Graph Linear Inequalities 1
Convert Standard Form and Slope Intercept Form 1
Convert Standard Form and Slope Intercept Form 2
Point Slope Form 1
Point Slope Form 2
Parallel Lines 1
Parallel Lines 2
Perpendicular Lines 1
Perpendicular Lines 2
Simplify Roots of Negative Numbers 1
Simplify Roots of Negative Numbers 2
Powers of the Imaginary Unit 1
Powers of the Imaginary Unit 2
Solve Quadratic Equations with Complex Solutions 1
Solve Quadratic Equations with Complex Solutions 2
Equality of Complex Numbers
Add and Subtract Complex Numbers 1
Add and Subtract Complex Numbers 2
Multiply Complex Numbers 1
Multiply Complex Numbers 2
Divide Complex Numbers
Complex Numbers – Product of Linear Factors 1
Complex Numbers – Product of Linear Factors 2
Mixed Operations with Complex Numbers
Solve a System of Equations by Graphing
Substitution Method 1
Substitution Method 2
Substitution Method 3
Substitution Method 4
Elimination Method 1
Elimination Method 2
Elimination Method 3
Elimination Method 4
Systems of Nonlinear Equations
Systems of Equations Word Problems 1
Systems of Equations Word Problems 2
3 Variable Systems of Equations – Substitution Method
3 Variable Systems of Equations – Elimination Method
Identify Functions
Function Notation
Domain and Range 1
Domain and Range 2
Domain and Range 3
Odd and Even Functions 1
Odd and Even Functions 2
Piecewise Functions
Graphing the Intersection of Regions 1
Graphing the Intersection of Regions 2
Tangents and Normals
Log and Exponential Tangents and Normals
Increasing and Decreasing Intervals
Relative and Absolute Maxima and Minima
Stationary and Inflection Points
Curve Sketching 1
Curve Sketching 2
Optimisation Problems
Trig Applications of Differentiation
Write an Equation for Circle Graphs
Graph Circles in Standard Form
Graph Circles in Expanded Form
Graph Parabolas (Focus and Directrix) 1
Graph Parabolas (Focus and Directrix) 2
Graph Parabolas Given the Vertex, Focus and Directrix 1
Graph Parabolas Given the Vertex, Focus and Directrix 2
Graph Hyperbolas
Write an Equation for Hyperbolas
Graph Cubic Curves
Write an Equation for Cubic Curves
The Exponential Curve
Write an Equation for Circle Graphs
Graph Circles in Standard Form
Graph Circles in Expanded Form
Graph Parabolas (Focus and Directrix) 1
Graph Parabolas (Focus and Directrix) 2
Graph Parabolas Given the Vertex, Focus and Directrix 1
Graph Parabolas Given the Vertex, Focus and Directrix 2
Graph Hyperbolas
Write an Equation for Hyperbolas
Graph Cubic Curves
Write an Equation for Cubic Curves
The Exponential Curve
Lowest Common Denominator
Add & Subtract Rational Expressions
Add & Subtract Rational Expressions with Unlike Denominators 1
Add & Subtract Rational Expressions with Unlike Denominators 2
Multiply & Divide Rational Expressions
Lowest Common Denominator
Add & Subtract Rational Expressions
Add & Subtract Rational Expressions with Unlike Denominators 1
Add & Subtract Rational Expressions with Unlike Denominators 2
Multiply & Divide Rational Expressions
Area of Shapes 1
Area of Shapes 2
Area of Shapes 3
Area of Shapes 4
Areas of Circles 1
Areas of Circles 2
Areas of a Shaded Region
Area of Composite Shapes
Area of Sectors
Area of Shapes Mixed Review 1
Area of Shapes Mixed Review 2
Area of Shapes Mixed Review 3
Area of Shapes 1
Area of Shapes 2
Area of Shapes 3
Area of Shapes 4
Areas of Circles 1
Areas of Circles 2
Areas of a Shaded Region
Area of Composite Shapes
Area of Sectors
Area of Shapes Mixed Review 1
Area of Shapes Mixed Review 2
Area of Shapes Mixed Review 3
Volume of Shapes 1
Volume of Shapes 2
Volume of Shapes 3
Volume of Shapes 4
Volume of Composite Shapes 1
Volume of Composite Shapes 2
Surface Area of Shapes 1
Surface Area of Shapes 2
Surface Area of Shapes 3
Surface Area and Volume Mixed Review 1
Surface Area and Volume Mixed Review 2
Surface Area and Volume Mixed Review 3
Surface Area and Volume Mixed Review 4
Greatest Common Factor 1
Greatest Common Factor 2
Factor Expressions using GCF
Factor Expressions 1
Factor Expressions 2
Factor Expressions with Negative Numbers
Factor Difference of Two Squares 1
Factor Difference of Two Squares 2
Factor Difference of Two Squares 3
Factor by Grouping
Factor Difference of Two Squares (Harder) 1
Factor Difference of Two Squares (Harder) 2
Factor Difference of Two Squares (Harder) 3
Factor Quadratics 1
Factor Quadratics 2
Factor Quadratics 3
Factor Quadratics with Leading Coefficient more than 1 (1)
Factor Quadratics with Leading Coefficient more than 1 (2)
Factor Quadratics with Leading Coefficient more than 1 (3)
Factor Quadratics (Complex)
Simple Probability 1
Simple Probability 2
Simple Probability 3
Simple Probability 4
Complementary Probability 1
Compound Events 1
Compound Events 2
Venn Diagrams (Non Mutually Exclusive)
Independent Events 1
Independent Events 2
Dependent Events (Conditional Probability)
Probability Tree (Independent) 1
Probability Tree (Independent) 2
Probability Tree (Dependent)
Simple Probability 1
Simple Probability 2
Simple Probability 3
Simple Probability 4
Complementary Probability 1
Compound Events 1
Compound Events 2
Venn Diagrams (Non Mutually Exclusive)
Independent Events 1
Independent Events 2
Dependent Events (Conditional Probability)
Find the Hypotenuse 1
Find the Hypotenuse 2
Find the Hypotenuse 3
Find a Side 1
Find a Side 2
Find a Side 3
Pythagoras’ Theorem Problems 1
Pythagoras’ Theorem Problems 2
Pythagoras’ Theorem Problems 3
Pythagoras Theorem Mixed Review 1
Pythagoras Theorem Mixed Review 2
Pythagoras Theorem Mixed Review 3
Pythagoras Theorem Mixed Review 4
Distance Between Two Points 1
Distance Between Two Points 2
Distance Between Two Points 3
Midpoint Formula 1
Midpoint Formula 2
Midpoint Formula 3
Parallel Lines 1
Parallel Lines 2
Perpendicular Lines 1
Perpendicular Lines 2
Solve Quadratics by Factoring 1
Solve Quadratics by Factoring 2
The Quadratic Formula
Completing the Square 1
Completing the Square 2
Intro to Quadratic Functions (Parabolas) 1
Intro to Quadratic Functions (Parabolas) 2
Intro to Quadratic Functions (Parabolas) 3
Graph Quadratic Functions in Standard Form 1
Graph Quadratic Functions in Standard Form 2
Graph Quadratic Functions by Completing the Square
Graph Quadratic Functions in Vertex Form
Write a Quadratic Equation from the Graph
Write a Quadratic Equation Given the Vertex and Another Point
Quadratic Inequalities 1
Quadratic Inequalities 2
Congruent Triangles 1
Congruent Triangles 2
Pythagoras’ Theorem Problems 1
Pythagoras’ Theorem Problems 2
Pythagoras’ Theorem Problems 3
Solve Quadratics by Factoring
The Quadratic Formula
Completing the Square 1
Completing the Square 2
Intro to Quadratic Functions (Parabolas) 1
Intro to Quadratic Functions (Parabolas) 2
Intro to Quadratic Functions (Parabolas) 3
Graph Quadratic Functions in Standard Form 1
Graph Quadratic Functions in Standard Form 2
Graph Quadratic Functions by Completing the Square
Graph Quadratic Functions in Vertex Form
Write a Quadratic Equation from the Graph
Write a Quadratic Equation Given the Vertex and Another Point
Quadratic Inequalities 1
Quadratic Inequalities 2
Quadratics Word Problems 1
Quadratics Word Problems 2
Quadratic Identities
Graphing Quadratics Using the Discriminant
Positive and Negative Definite
Applications of the Discriminant 1
Applications of the Discriminant 2
Combining Methods for Solving Quadratic Equations
Solve Quadratics by Factoring
The Quadratic Formula
Completing the Square 1
Completing the Square 2
Intro to Quadratic Functions (Parabolas) 1
Intro to Quadratic Functions (Parabolas) 2
Intro to Quadratic Functions (Parabolas) 3
Graph Quadratic Functions in Standard Form 1
Graph Quadratic Functions in Standard Form 2
Graph Quadratic Functions by Completing the Square
Graph Quadratic Functions in Vertex Form
Write a Quadratic Equation from the Graph
Write a Quadratic Equation Given the Vertex and Another Point
Quadratic Inequalities 1
Quadratic Inequalities 2
Quadratics Word Problems 1
Quadratics Word Problems 2
Quadratic Identities
Graphing Quadratics Using the Discriminant
Positive and Negative Definite
Applications of the Discriminant 1
Applications of the Discriminant 2
Combining Methods for Solving Quadratic Equations
Power Rule 1
Power Rule 2
Power Rule 3
Power Rule 4
Chain Rule 1
Chain Rule 2
Product Rule
Quotient Rule
Derivatives of Exponential Functions 1
Derivatives of Exponential Functions 2
Derivatives of Exponential Functions 3
Derivatives of Trigonometric Functions 1
Derivatives of Trigonometric Functions 2
Derivatives of Trigonometric Functions 3
Power Rule 1
Power Rule 2
Power Rule 3
Power Rule 4
Chain Rule 1
Chain Rule 2
Product Rule
Quotient Rule
Derivatives of Exponential Functions 1
Derivatives of Exponential Functions 2
Derivatives of Exponential Functions 3
Derivatives of Trigonometric Functions 1
Derivatives of Trigonometric Functions 2
Derivatives of Trigonometric Functions 3
Tangents and Normals
Log and Exponential Tangents and Normals
Increasing and Decreasing Intervals
Critical Points (Maximum and Minimum Values)
Stationary and Inflection Points
Curve Sketching 1
Curve Sketching 2
Optimization
Tangents and Normals
Log and Exponential Tangents and Normals
Increasing and Decreasing Intervals
Critical Points (Maximum and Minimum Values)
Stationary and Inflection Points
Curve Sketching 1
Curve Sketching 2
Optimization
Antiderivatives (Indefinite Integrals) 1
Antiderivatives (Indefinite Integrals) 2
Antiderivatives (Indefinite Integrals) 3
Antiderivatives of Exponential Functions
Antiderivatives of Logarithmic Functions 1
Antiderivatives of Logarithmic Functions 2
Antiderivatives of Trig Functions 1
Antiderivatives of Trig Functions 2
Definite Integrals
Definite Integrals of Exponential Functions
Definite Integrals of Logarithmic Functions
Antiderivatives (Indefinite Integrals) 1
Antiderivatives (Indefinite Integrals) 2
Antiderivatives (Indefinite Integrals) 3
Antiderivatives of Exponential Functions
Antiderivatives of Logarithmic Functions 1
Antiderivatives of Logarithmic Functions 2
Antiderivatives of Trig Functions 1
Antiderivatives of Trig Functions 2
Definite Integrals
Definite Integrals of Exponential Functions
Definite Integrals of Logarithmic Functions
Areas Between Curves and the Axis 1
Areas Between Curves and the Axis 2
Area Between Two Curves
Volumes of Solids 1
Volumes of Solids 2
Volumes of Solids 3
Areas Between Curves and the Axis 1
Areas Between Curves and the Axis 2
Area Between Two Curves
Volumes of Solids 1
Volumes of Solids 2
Volumes of Solids 3
Intro to Sequences
Arithmetic Series (Sum)
Geometric Series (Sum)
Infinite Geometric Series