Graphing Reflections 1
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Question 1 of 8
1. Question
Given `f(x)=x^2-4x`
Sketch `y=-f(x)`.
Correct
Great Work!
Incorrect
Reflections around the `x`-axis have the property `\ \ y=-f(x)`Replace `y\rightarrow-y`.To sketch the transformed function `y=-f(x)`, start by sketching the original function `f(x)=x^2-4x`.Find the points where the graph of `f(x)=x^2-4x` crosses the `x`-axis.`0=` `x^2-4x` Set `y=0` to find the points where the graph crosses the `x`-axis. `0=` `color(green)(x)(x-4)` Factor out `color(green)(x)` from both terms. Solve `x=0`.`x=` `0` `y=` `(0)^2-4(0)` When `x=0`, solve for `y`. `y=` `0` `A(0,0)` Solve `x-4=0`.`x-4=` `0` `x=` `4` `y=` `(4)^2-4(4)` When `x=4`, solve for `y`. `y=` `16-16` `y=` `0` `B(4,0)` Find the vertex. Take the two `x` values and find the halfway point `((4+0)/2=2)`.`y=` `(2)^2-4(2)` Find the `y` value when `x=2`. `y=` `4-8` `y=` `-4` Vertex 1`(2,-4)` Sketch the original function `f(x)=x^2-4x` by using the points Vertex`(2,-4)`, `A(0,0)` and `B(4,0)`. Remember that this function is a concave up parabola (positive `x^2` function).
Now sketch the transformed function `y=-f(x)` which is `f(x)=-(x^2-4x)=-x^2+4x`.Find the points where the graph of `f(x)=-x^2+4x` crosses the `x`-axis.`0=` `-x^2+4x` Set `y=0` to find the points where the graph crosses the `x`-axis. `0=` `-x(x-4)` Factor out a `-x` from both terms. Solve `-x=0`.`-x=` `0` `x=` `0` `y=` `(0)^2-4(0)` When `x=0`, solve for `y`. `y=` `0` `C(0,0)` Solve `x-4=0`.`x-4=` `0` `x=` `4` `y=` `(4)^2-4(4)` When `x=4`, solve for `y`. `y=` `16-16` `y=` `0` `D(4,0)` Find the vertex. Take the two `x` values and find the halfway point `((4+0)/2=2)`.`y=` `-(2)^2+4(2)` Find the `y` value when `x=2`. `y=` `-4+8` `y=` `4` Vertex 2`(2,4)` Sketch the transformed function `f(x)=-x^2+4x` by using the points Vertex 2`(2,4)`, `C(0,0)` and `D(4,0)`. Remember that this function is a concave down parabola (negative`x^2` function).
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Question 2 of 8
2. Question
Graph `y=(x-3)^2` reflected in the `x`-axis.
Correct
Great Work!
Incorrect
Reflections around the `x`-axis have the property `\ \ y=-f(x)`Replace `y\rightarrow-y`.To sketch the reflected function, take `y=(x-3)^2` and then replace `y` for `-y` and you will get: `y=-(x-3)^2`.Use a table of values to sketch `y=(x-3)^2` and `y=-(x-3)^2`.`x` `1` `2` `3` `4` `5` `(x-3)^2` `4` `1` `0` `1` `4` `-(x-3)^2` `-4` `-1` `0` `-1` `-4` Sketch the graph of `y=(x-3)^2` and `y=-(x-3)^2` using the table of values.
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Question 3 of 8
3. Question
Graph `y=2^x` reflected in the `y`-axis.
Correct
Great Work!
Incorrect
Reflections around the `y`-axis have the property `\ \ y=f(-x)`Replace `x\rightarrow-x`.To sketch the reflected function, take `y=2^x` and then
replace `x` for `-x` and you will get: `y=2^(-x)`.Use a table of values to sketch `y=2^x` and `y=2^-x`.`x` `-3` `-2` `-1` `0` `1` `2` `3` `2^x` `1/8` `1/4` `1/2` `1` `2` `4` `8` `2^(-x)` `8` `4` `2` `1` `1/2` `1/4` `1/8` Sketch the graph of `y=2^x` and `y=2^(-x)` using the table of values.
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Question 4 of 8
4. Question
Graph `y=(x-1)^2` reflected in the `y`-axis.
Correct
Great Work!
Incorrect
Reflections around the `y`-axis have the property `\ \ y=f(-x)`Replace `x\rightarrow-x`.To sketch the reflected function, take `y=(x-1)^2` and then replace `x` for `-x` and you will get: `y=(-x-1)^2`.Use a table of values to sketch `y=(x-1)^2` and `y=(-x-1)^2` which can also be written as:`y=(-1)^2` `(-x-1)^2 \rightarrow y=(x+1)^2``x` `-2` `-1` `0` `1` `2` `3` `(x-1)^2` `9` `4` `1` `0` `1` `4` `(x+1)^2` `1` `0` `1` `4` `9` `16` Sketch the graph of `y=(x-1)^2` and `y=(x+1)^2` using the table of values.
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Question 5 of 8
5. Question
Graph `y=\sqrt(x)` reflected in the `x`-axis.
Correct
Great Work!
Incorrect
Reflections around the `x`-axis have the property `\ \ y=-f(x)`Replace `y\rightarrow-y`.To sketch the reflected function, take `y=\sqrt(x)` and then
replace `y` for `-y` and you will get: `y=-\sqrt(x)`.Use a table of values to sketch `y=\sqrt(x)` and `y=-\sqrt(x)`.`x` `0` `1` `2` `3` `4` `5` `\sqrt(x)` `0` `1` `1.4` `1.7` `2` `2.2` `-\sqrt(x)` `0` `-1` `-1.4` `-1.7` `-2` `-2.2` Sketch the graph of `y=\sqrt(x)` and `y=-\sqrt(x)` using the table of values.
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Question 6 of 8
6. Question
Given `f(x)=x^3`
Sketch `y=-f(x)`
Correct
Great Work!
Incorrect
Reflections around the `x`-axis have the property `\ \ y=-f(x)`Replace `y\rightarrow-y`.To sketch the reflected function, take `y=x^3` and then
replace `y` for `-y` and you will get: `y=-x^3`Use a table of values to sketch `y=x^3` and `y=-x^3``x` `-2` `-1` `0` `1` `2` `x^3` `-8` `-1` `0` `1` `8` `-x^3` `8` `1` `0` `-1` `-8` Sketch the graph of `y=x^3` and `y=-x^3` using the table of values.
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Question 7 of 8
7. Question
Given `f(x)=x^3 +1`.
Sketch `y=f(-x)`
Correct
Great Work!
Incorrect
Reflections around the `y`-axis have the property `\ \ y=f(-x)`Replace `x\rightarrow-x`.To sketch the transformed function `y=f(-x)`, start by sketching the original function `f(x)=x^3 +1`.Find the points where the graph of `f(x)=x^3 +1` crosses the `y`-axis.Solve `x=0`.`x=` `0` `y=` `(0)^3 +1` When `x=0`, solve for `y`. `y=` `1` `A(0,1)` Solve `x^3 + 1=0`.`x^3 + 1=` `0` `x=` `-1` `y=` `(-1)^3 + 1` When `x=-1`, solve for `y`. `y=` `-1+1` `y=` `0` `B(-1,0)` Sketch the original function `f(x)=x^3 +1` by using the points`A(0,1)` and `B(-1,0)`. Remember that this function is a cubic function (positive `x^3` function).
Now sketch the transformed function `y=f(-x)` which is `f(-x)=(-x)^3 + 1=-x^3 + 1`.Find the points where the graph of `f(-x)=-x^3 + 1` crosses the `y`-axis.Solve `-x=0`.`-x=` `0` `x=` `0` `y=` `(0)^3 + 1` When `x=0`, solve for `y`. `y=` `1` `C(0,1)` Solve `-x^3 + 1=0`.`-x^3 + 1=` `0` `x=` `1` `y=` `-(1)^3 + 1` When `x=1`, solve for `y`. `y=` `-1+1` `y=` `0` `D(1,0)` Sketch the transformed function `f(-x)=-x^3+1` by using the points `C(0,1)` and `D(1,0)`. Remember that this is a negative cubic function.
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Question 8 of 8
8. Question
Given `f(x)=x^2+4x`.
Sketch `y=f(-x)`
Correct
Great Work!
Incorrect
Reflections around the `y`-axis have the property `\ \ y=f(-x)`.Replace `x\rightarrow-x`.To sketch the transformed function `y=f(-x)`, start by sketching the original function `f(x)=x^2 + 4x`.Find the points where the graph of `f(x)=x^2+4x` crosses the `y`-axis.`0=` `x^2+4x` Set `y=0` to find the points where the graph crosses the `x`-axis. `0=` `color(green)(x)(x+4)` Factor out an `color(green)(x)` from both terms. Solve `x=0`.`x=` `0` `y=` `(0)^2+4(0)` When `x=0`, solve for `y`. `y=` `0` `A(0,0)` Solve `x+4=0`.`x+4=` `0` `x=` `-4` `y=` `(-4)^2+4(-4)` When `x=-4`, solve for `y`. `y=` `16-16` `y=` `0` `B(-4,0)` Find the vertex. Take the two `x` values and find the halfway point `((-4+0)/2=-2)`.`y=` `(-2)^2+4(-2)` Find the `y` value when `x=-2`. `y=` `4-8` `y=` `-4` Vertex 1`(-2,-4)` Sketch the original function `f(x)=x^2+4x` by using the points Vertex`(-2,-4)`, `A(0,0)` and `B(-4,0)`. Remember that this function is a concave up parabola (positive `x^2` function).
Now sketch the transformed function `y=f(-x)` which is `f(-x)=(-x)^2+4(-x)=x^2-4x`.Find the points where the graph of `f(-x)=x^2-4x` crosses the `x`-axis.`0=` `x^2-4x` Set `y=0` to find the points where the graph crosses the `x`-axis. `0=` `color(green)(x)(x-4)` Factor out an `color(green)(x)` from both terms. Solve `x=0`.`x=` `0` `y=` `(0)^2-4(0)` When `x=0`, solve for `y`. `y=` `0` `C(0,0)` Solve `x-4=0`.`x-4=` `0` `x=` `4` `y=` `(4)^2-4(4)` When `x=4`, solve for `y`. `y=` `16-16` `y=` `0` `D(4,0)` Find the vertex. Take the two `x` values and find the halfway point `((4+0)/2=2)`.`y=` `(2)^2-4(2)` Find the `y` value when `x=2`. `y=` `4-8` `y=` `-4` Vertex 2`(2,-4)` Sketch the transformed function `f(-x)=x^2-4x` by using the points Vertex 2 `(2,-4)`, `C(0,0)` and `D(4,0)`. Remember that this function is a concave up parabola (positive `x^2` function).
Quizzes
- 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