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Graphing Quadratics Using the DiscriminantGraphing Quadratics Using the Discriminant
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Question 1 of 7
1. Question
Which of the following graphs have a discriminant that is equal to `0`?
(`Delta=0`)Hint
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Nature of the Roots Discriminant (`Delta`) Two real roots `Delta``>``0` One real root `Delta=0` No real roots `Delta``<``0` Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$For each graph, check how many times the graph has passed through the `x` axis
This graph touched the `x` axis once, which means it has one rootHence, `Delta=0`
This graph touched the `x` axis twice, which means it has two rootsHence, `Delta``>``0`
This graph touched the `x` axis once, which means it has one rootHence, `Delta=0`
This graph touched the `x` axis twice, which means it has two rootsHence, `Delta``>``0`
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Question 2 of 7
2. Question
Which function does not intersect with the `x` axis?Hint
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Nature of the Roots Discriminant (`Delta`) Two real roots `Delta``>``0` One real root `Delta=0` No real roots `Delta``<``0` Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$Compute for the discriminant of each function`x^2-4x+3=0``a=1` `b=-4` `c=3``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{(-4)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(3)}$$ Substitute values `=` `16-12` `=` `4` Since `Delta``>``0`, this function has two real roots and intersects the `x` axis twice`x^2-4x+4=0``a=1` `b=-4` `c=4``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{(-4)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(4)}$$ Substitute values `=` `16-16` `=` `0` Since `Delta=0`, this function has one real root and intersects the `x` axis once`x^2-4x+5=0``a=1` `b=-4` `c=5``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{(-4)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(5)}$$ Substitute values `=` `16-20` `=` `-4` Since `Delta``<``0`, this function has no real roots and does not intersect the `x` axis`x^2-4x+5=0` -
Question 3 of 7
3. Question
Graph using the discriminant`y=2x^2-2x-5`Hint
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Quadratic Formula
$$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, compute for the discriminant`y=2x^2-2x-5``a=2` `b=-2` `c=-5``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{-2}}^2-4\color{#00880A}{(2)}\color{#007DDC}{(-5)}$$ Substitute values `=` `4+40` `=` `44` Next, substitute the discriminant to the Quadratic Formula to find the `x` intercepts`x` `=` $$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$ Quadratic Formula `=` $$\frac {-\color{#9a00c7}{(-2)} \pm \sqrt {44} }{2 \color{#00880A}{(2)}}$$ Substitute values `=` `(2+-2sqrt11)/4` `=` `(1+-sqrt11)/2` Write each root individually$$x_1$$ `=` `(1+sqrt11)/2` `=` $$2.158$$ $$x_1$$ `=` `(1-sqrt11)/2` `=` $$-1.158$$ Mark these `2` points on the `x` axis
Find the `y` intercept by substituting `x=0``y` `=` `2x^2-2x-5` `=` `2(0)^2-2(0)-5` Substitute `x=0` `=` `0-0-5` `=` `-5` Mark this on the `y` axis
Finally, form the parabola by connecting the points
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Question 4 of 7
4. Question
Graph using the discriminant`y=3x^2-4x-4`Hint
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Quadratic Formula
$$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, compute for the discriminant`y=3x^2-4x-4``a=3` `b=-4` `c=-4``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{-4}}^2-4\color{#00880A}{(3)}\color{#007DDC}{(-4)}$$ Substitute values `=` `16+48` `=` `64` Next, substitute the discriminant to the Quadratic Formula to find the `x` intercepts`x` `=` $$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$ Quadratic Formula `=` $$\frac {-\color{#9a00c7}{(-4)} \pm \sqrt {64} }{2 \color{#00880A}{(3)}}$$ Substitute values `=` `(4+-8)/6` Write each root individually$$x_1$$ `=` `(4+8)/6` `=` `12/6` `=` `2` $$x_1$$ `=` `(4-8)/6` `=` `(-4)/6` `=` `-2/3` Mark these `2` points on the `x` axis
Find the `y` intercept by substituting `x=0``y` `=` `3x^2-4x-4` `=` `3(0)^2-4(0)-4` Substitute `x=0` `=` `0-0-4` `=` `-4` Mark this on the `y` axis
Finally, form the parabola by connecting the points
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Question 5 of 7
5. Question
Graph using the discriminant`y=2x^2+3x-1`Hint
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Quadratic Formula
$$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, compute for the discriminant`y=2x^2+3x-1``a=2` `b=3` `c=-1``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{3}}^2-4\color{#00880A}{(2)}\color{#007DDC}{(-1)}$$ Substitute values `=` `9+8` `=` `17` Next, substitute the discriminant to the Quadratic Formula to find the `x` intercepts`x` `=` $$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$ Quadratic Formula `=` $$\frac {-\color{#9a00c7}{3} \pm \sqrt {17} }{2 \color{#00880A}{(2)}}$$ Substitute values `=` `(-3+-sqrt17)/4` Write each root individually$$x_1$$ `=` `(-3+sqrt17)/4` `=` `1.123/4` `=` `0.281` $$x_1$$ `=` `(-3-sqrt17)/4` `=` `(-7.123)/4` `=` `-1.781` Mark these `2` points on the `x` axis
Find the `y` intercept by substituting `x=0``y` `=` `2x^2+3x-1` `=` `2(0)^2+3(0)-1` Substitute `x=0` `=` `0+0-1` `=` `-1` Mark this on the `y` axis
Finally, form the parabola by connecting the points
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Question 6 of 7
6. Question
Graph using the discriminant`y=x^2-6x+9`Hint
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Quadratic Formula
$$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$Axis of Symmetry
$$x=\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$$Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, find the axis of symmetry`y=x^2-6x+9``a=1` `b=-6` `c=9``x` `=` $$\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$$ Axis of Symmetry `x` `=` $$\frac{-\color{#9a00c7}{(-6)}}{2\color{#00880A}{(1)}}$$ Substitute values `x` `=` `6/2` `x` `=` `3` 
Find where the graph touches the axis of symmetry by substituting `x=3` to the function. This would be the vertex`y` `=` `x^2-6x+9` `=` `(3)^2-6(3)+9` Substitute `x=3` `=` `9-18+9` `=` `0` Hence, the vertex is at `(3,0)`
Next, compute for the discriminant`Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{-6}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(9)}$$ Substitute values `=` `36-36` `=` `0` Substitute the discriminant to the Quadratic Formula to find the `x` intercepts`x` `=` $$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$ Quadratic Formula `=` $$\frac {-\color{#9a00c7}{(-6)} \pm \sqrt {0} }{2 \color{#00880A}{(1)}}$$ Substitute values `=` `(6+-0)/2` `=` `(6)/2` `=` `3` There is only one root or `x` intercept which is at `(3,0)`Recall that the `(3,0)` is also the vertexFind the `y` intercept by substituting `x=0``y` `=` `x^2-6x+9` `=` `(0)^2-6(0)+9` Substitute `x=0` `=` `0-0+9` `=` `9` Mark this on the `y` axis
Finally, form the parabola by connecting the points
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Question 7 of 7
7. Question
Graph using the discriminant`y=x^2+2x-8`Hint
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Quadratic Formula
$$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$Axis of Symmetry
$$x=\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$$Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, find the axis of symmetry`y=x^2+2x-8``a=1` `b=2` `c=-8``x` `=` $$\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$$ Axis of Symmetry `x` `=` $$\frac{-\color{#9a00c7}{2}}{2\color{#00880A}{(1)}}$$ Substitute values `x` `=` `-2/2` `x` `=` `-1` 
Find where the graph touches the axis of symmetry by substituting `x=3` to the function. This would be the vertex`y` `=` `x^2+2x-8` `=` `(-1)^2+2(-1)-8` Substitute `x=-1` `=` `1-2-8` `=` `-9` Hence, the vertex is at `(-1,-9)`
Next, compute for the discriminant`Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{-2}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(-8)}$$ Substitute values `=` `4+32` `=` `36` Substitute the discriminant to the Quadratic Formula to find the `x` intercepts`x` `=` $$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$$ Quadratic Formula `=` $$\frac {-\color{#9a00c7}{2} \pm \sqrt {36} }{2 \color{#00880A}{(1)}}$$ Substitute values `=` `(-2+-6)/2` `=` `-1+-3` `=` `-1-3,-1+3` `x` `=` `-4,2` 
Finally, form the parabola by connecting the points
Quizzes
- 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