Graphing Quadratics Using the Discriminant

Question 1 of 7

Which of the following graphs have a discriminant that is equal to

0

$0$ ?
(

Δ = 0

$Delta=0$ )

1.
2.
3.
4.

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Nature of the Roots	Discriminant ( $Δ$ $Delta$ )
Two real roots	$Δ$ $Delta$ $>$ $>$ $0$ $0$
One real root	$Δ = 0$ $Delta=0$
No real roots	$Δ$ $Delta$ $<$ $<$ $0$ $0$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

For each graph, check how many times the graph has passed through the

x

$x$ axis

This graph touched the

x

$x$ axis once, which means it has one root

Hence,

Δ = 0

$Delta=0$

This graph touched the

x

$x$ axis twice, which means it has two roots

Hence,

Δ

$Delta$

>

$>$

0

$0$

This graph touched the

x

$x$ axis once, which means it has one root

Hence,

Δ = 0

$Delta=0$

This graph touched the

x

$x$ axis twice, which means it has two roots

Hence,

Δ

$Delta$

>

$>$

0

$0$

Question 2 of 7

Which function does not intersect with the

x

$x$ axis?

1. $x^{2} - 4 x + 4 = 0$ $x^2-4x+4=0$
2. $x^{2} - 4 x + 5 = 0$ $x^2-4x+5=0$
3. $x^{2} - 4 x + 3 = 0$ $x^2-4x+3=0$

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Nature of the Roots	Discriminant ( $Δ$ $Delta$ )
Two real roots	$Δ$ $Delta$ $>$ $>$ $0$ $0$
One real root	$Δ = 0$ $Delta=0$
No real roots	$Δ$ $Delta$ $<$ $<$ $0$ $0$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

Compute for the discriminant of each function

x^{2} - 4 x + 3 = 0

$x^2-4x+3=0$

$a = 1$ $a=1$ $b = - 4$ $b=-4$ $c = 3$ $c=3$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{(-4)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(3)}$	Substitute values
	$=$ $=$	$16 - 12$ $16-12$
	$=$ $=$	$4$ $4$

Since

Δ

$Delta$

>

$>$

0

$0$ , this function has two real roots and intersects the

x

$x$ axis twice

x^{2} - 4 x + 4 = 0

$x^2-4x+4=0$

$a = 1$ $a=1$ $b = - 4$ $b=-4$ $c = 4$ $c=4$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{(-4)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(4)}$	Substitute values
	$=$ $=$	$16 - 16$ $16-16$
	$=$ $=$	$0$ $0$

Since

Δ = 0

$Delta=0$ , this function has one real root and intersects the

x

$x$ axis once

x^{2} - 4 x + 5 = 0

$x^2-4x+5=0$

$a = 1$ $a=1$ $b = - 4$ $b=-4$ $c = 5$ $c=5$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{(-4)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(5)}$	Substitute values
	$=$ $=$	$16 - 20$ $16-20$
	$=$ $=$	$- 4$ $-4$

Since

Δ

$Delta$

<

$<$

0

$0$ , this function has no real roots and does not intersect the

x

$x$ axis

x^{2} - 4 x + 5 = 0

$x^2-4x+5=0$

Question 3 of 7

Graph using the discriminant

y = 2 x^{2} - 2 x - 5

$y=2x^2-2x-5$

1.
2.
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4.

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Quadratic Formula

x = \frac{- b \pm \sqrt{Δ}}{2 a}

$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, compute for the discriminant

y = 2 x^{2} - 2 x - 5

$y=2x^2-2x-5$

$a = 2$ $a=2$ $b = - 2$ $b=-2$ $c = - 5$ $c=-5$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{-2}}^2-4\color{#00880A}{(2)}\color{#007DDC}{(-5)}$	Substitute values
	$=$ $=$	$4 + 40$ $4+40$
	$=$ $=$	$44$ $44$

Next, substitute the discriminant to the Quadratic Formula to find the

x

$x$ intercepts

$x$ $x$	$=$ $=$	$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$	Quadratic Formula

	$=$ $=$	$\frac {-\color{#9a00c7}{(-2)} \pm \sqrt {44} }{2 \color{#00880A}{(2)}}$	Substitute values

	$=$ $=$	$\frac{2 \pm 2 \sqrt{11}}{4}$ $(2+-2sqrt11)/4$

	$=$ $=$	$\frac{1 \pm \sqrt{11}}{2}$ $(1+-sqrt11)/2$

Write each root individually

$x_1$	$=$ $=$	$\frac{1 + \sqrt{11}}{2}$ $(1+sqrt11)/2$

	$=$ $=$	$2.158$

$x_1$	$=$ $=$	$\frac{1 - \sqrt{11}}{2}$ $(1-sqrt11)/2$

	$=$ $=$	$-1.158$

Mark these

2

$2$ points on the

x

$x$ axis

Find the

y

$y$ intercept by substituting

x = 0

$x=0$

$y$ $y$	$=$ $=$	$2 x^{2} - 2 x - 5$ $2x^2-2x-5$
	$=$ $=$	$2 {(0)}^{2} - 2 (0) - 5$ $2(0)^2-2(0)-5$	Substitute $x = 0$ $x=0$
	$=$ $=$	$0 - 0 - 5$ $0-0-5$
	$=$ $=$	$- 5$ $-5$

Mark this on the

y

$y$ axis

Finally, form the parabola by connecting the points

Question 4 of 7

Graph using the discriminant

y = 3 x^{2} - 4 x - 4

$y=3x^2-4x-4$

1.
2.
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4.

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Quadratic Formula

x = \frac{- b \pm \sqrt{Δ}}{2 a}

$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, compute for the discriminant

y = 3 x^{2} - 4 x - 4

$y=3x^2-4x-4$

$a = 3$ $a=3$ $b = - 4$ $b=-4$ $c = - 4$ $c=-4$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{-4}}^2-4\color{#00880A}{(3)}\color{#007DDC}{(-4)}$	Substitute values
	$=$ $=$	$16 + 48$ $16+48$
	$=$ $=$	$64$ $64$

Next, substitute the discriminant to the Quadratic Formula to find the

x

$x$ intercepts

$x$ $x$	$=$ $=$	$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$	Quadratic Formula

	$=$ $=$	$\frac {-\color{#9a00c7}{(-4)} \pm \sqrt {64} }{2 \color{#00880A}{(3)}}$	Substitute values

	$=$ $=$	$\frac{4 \pm 8}{6}$ $(4+-8)/6$

Write each root individually

$x_1$	$=$ $=$	$\frac{4 + 8}{6}$ $(4+8)/6$

	$=$ $=$	$\frac{12}{6}$ $12/6$

	$=$ $=$	$2$ $2$

$x_1$	$=$ $=$	$\frac{4 - 8}{6}$ $(4-8)/6$

	$=$ $=$	$\frac{- 4}{6}$ $(-4)/6$

	$=$ $=$	$- \frac{2}{3}$ $-2/3$

Mark these

2

$2$ points on the

x

$x$ axis

Find the

y

$y$ intercept by substituting

x = 0

$x=0$

$y$ $y$	$=$ $=$	$3 x^{2} - 4 x - 4$ $3x^2-4x-4$
	$=$ $=$	$3 {(0)}^{2} - 4 (0) - 4$ $3(0)^2-4(0)-4$	Substitute $x = 0$ $x=0$
	$=$ $=$	$0 - 0 - 4$ $0-0-4$
	$=$ $=$	$- 4$ $-4$

Mark this on the

y

$y$ axis

Finally, form the parabola by connecting the points

Question 5 of 7

Graph using the discriminant

y = 2 x^{2} + 3 x - 1

$y=2x^2+3x-1$

1.
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Quadratic Formula

x = \frac{- b \pm \sqrt{Δ}}{2 a}

$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, compute for the discriminant

y = 2 x^{2} + 3 x - 1

$y=2x^2+3x-1$

$a = 2$ $a=2$ $b = 3$ $b=3$ $c = - 1$ $c=-1$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{3}}^2-4\color{#00880A}{(2)}\color{#007DDC}{(-1)}$	Substitute values
	$=$ $=$	$9 + 8$ $9+8$
	$=$ $=$	$17$ $17$

Next, substitute the discriminant to the Quadratic Formula to find the

x

$x$ intercepts

$x$ $x$	$=$ $=$	$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$	Quadratic Formula

	$=$ $=$	$\frac {-\color{#9a00c7}{3} \pm \sqrt {17} }{2 \color{#00880A}{(2)}}$	Substitute values

	$=$ $=$	$\frac{- 3 \pm \sqrt{17}}{4}$ $(-3+-sqrt17)/4$

Write each root individually

$x_1$	$=$ $=$	$\frac{- 3 + \sqrt{17}}{4}$ $(-3+sqrt17)/4$

	$=$ $=$	$\frac{1.123}{4}$ $1.123/4$

	$=$ $=$	$0.281$ $0.281$

$x_1$	$=$ $=$	$\frac{- 3 - \sqrt{17}}{4}$ $(-3-sqrt17)/4$

	$=$ $=$	$\frac{- 7.123}{4}$ $(-7.123)/4$

	$=$ $=$	$- 1.781$ $-1.781$

Mark these

2

$2$ points on the

x

$x$ axis

Find the

y

$y$ intercept by substituting

x = 0

$x=0$

$y$ $y$	$=$ $=$	$2 x^{2} + 3 x - 1$ $2x^2+3x-1$
	$=$ $=$	$2 {(0)}^{2} + 3 (0) - 1$ $2(0)^2+3(0)-1$	Substitute $x = 0$ $x=0$
	$=$ $=$	$0 + 0 - 1$ $0+0-1$
	$=$ $=$	$- 1$ $-1$

Mark this on the

y

$y$ axis

Finally, form the parabola by connecting the points

Question 6 of 7

Graph using the discriminant

y = x^{2} - 6 x + 9

$y=x^2-6x+9$

1.
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Quadratic Formula

x = \frac{- b \pm \sqrt{Δ}}{2 a}

$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$

Axis of Symmetry

x = \frac{- b}{2 a}

$x=\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, find the axis of symmetry

y = x^{2} - 6 x + 9

$y=x^2-6x+9$

$a = 1$ $a=1$ $b = - 6$ $b=-6$ $c = 9$ $c=9$

$x$ $x$	$=$ $=$	$\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$	Axis of Symmetry

$x$ $x$	$=$ $=$	$\frac{-\color{#9a00c7}{(-6)}}{2\color{#00880A}{(1)}}$	Substitute values

$x$ $x$	$=$ $=$	$\frac{6}{2}$ $6/2$

$x$ $x$	$=$ $=$	$3$ $3$

Find where the graph touches the axis of symmetry by substituting

x = 3

$x=3$ to the function. This would be the vertex

$y$ $y$	$=$ $=$	$x^{2} - 6 x + 9$ $x^2-6x+9$
	$=$ $=$	${(3)}^{2} - 6 (3) + 9$ $(3)^2-6(3)+9$	Substitute $x = 3$ $x=3$
	$=$ $=$	$9 - 18 + 9$ $9-18+9$
	$=$ $=$	$0$ $0$

Hence, the vertex is at

(3, 0)

$(3,0)$

Next, compute for the discriminant

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{-6}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(9)}$	Substitute values
	$=$ $=$	$36 - 36$ $36-36$
	$=$ $=$	$0$ $0$

Substitute the discriminant to the Quadratic Formula to find the

x

$x$ intercepts

$x$ $x$	$=$ $=$	$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$	Quadratic Formula

	$=$ $=$	$\frac {-\color{#9a00c7}{(-6)} \pm \sqrt {0} }{2 \color{#00880A}{(1)}}$	Substitute values

	$=$ $=$	$\frac{6 \pm 0}{2}$ $(6+-0)/2$

	$=$ $=$	$\frac{6}{2}$ $(6)/2$

	$=$ $=$	$3$ $3$

There is only one root or

x

$x$ intercept which is at

(3, 0)

$(3,0)$

Recall that the

(3, 0)

$(3,0)$ is also the vertex

Find the

y

$y$ intercept by substituting

x = 0

$x=0$

$y$ $y$	$=$ $=$	$x^{2} - 6 x + 9$ $x^2-6x+9$
	$=$ $=$	${(0)}^{2} - 6 (0) + 9$ $(0)^2-6(0)+9$	Substitute $x = 0$ $x=0$
	$=$ $=$	$0 - 0 + 9$ $0-0+9$
	$=$ $=$	$9$ $9$

Mark this on the

y

$y$ axis

Finally, form the parabola by connecting the points

Question 7 of 7

Graph using the discriminant

y = x^{2} + 2 x - 8

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

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2.
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4.

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Quadratic Formula

x = \frac{- b \pm \sqrt{Δ}}{2 a}

$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$

Axis of Symmetry

x = \frac{- b}{2 a}

$x=\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, find the axis of symmetry

y = x^{2} + 2 x - 8

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

$a = 1$ $a=1$ $b = 2$ $b=2$ $c = - 8$ $c=-8$

$x$ $x$	$=$ $=$	$\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$	Axis of Symmetry

$x$ $x$	$=$ $=$	$\frac{-\color{#9a00c7}{2}}{2\color{#00880A}{(1)}}$	Substitute values

$x$ $x$	$=$ $=$	$- \frac{2}{2}$ $-2/2$

$x$ $x$	$=$ $=$	$- 1$ $-1$

Find where the graph touches the axis of symmetry by substituting

x = 3

$x=3$ to the function. This would be the vertex

$y$ $y$	$=$ $=$	$x^{2} + 2 x - 8$ $x^2+2x-8$
	$=$ $=$	${(- 1)}^{2} + 2 (- 1) - 8$ $(-1)^2+2(-1)-8$	Substitute $x = - 1$ $x=-1$
	$=$ $=$	$1 - 2 - 8$ $1-2-8$
	$=$ $=$	$- 9$ $-9$

Hence, the vertex is at

(- 1, - 9)

$(-1,-9)$

Next, compute for the discriminant

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{-2}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(-8)}$	Substitute values
	$=$ $=$	$4 + 32$ $4+32$
	$=$ $=$	$36$ $36$

Substitute the discriminant to the Quadratic Formula to find the

x

$x$ intercepts

$x$ $x$	$=$ $=$	$\frac {-\color{#9a00c7}{b} \pm \sqrt {\Delta} }{2 \color{#00880A}{a}}$	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

Graphing Quadratics Using the Discriminant

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Quiz summary

Information

1. Question

Hint

Fantastic!

Discriminant Formula

2. Question

Hint

Nice Job!

Discriminant Formula

3. Question

Hint

Excellent!

Quadratic Formula

Discriminant Formula

4. Question

Hint

Fantastic!

Quadratic Formula

Discriminant Formula

5. Question

Hint

Keep Going!

Quadratic Formula

Discriminant Formula

6. Question

Hint

Fantastic!

Quadratic Formula

Axis of Symmetry

Discriminant Formula

7. Question

Hint

Keep Going!

Quadratic Formula

Axis of Symmetry

Discriminant Formula

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