Write a Parabola from the Graph

Question 1 of 7

Find the equation of the graph below

1. $y = 3 (x + 1) (x + 4)$ $y=3(x+1)(x+4)$
2. $y = (x - 1) (x - 4)$ $y=(x-1)(x-4)$
3. $y = (x + 1) (x - 4)$ $y=(x+1)(x-4)$
4. $y = 2 (x - 1) (x - 4)$ $y=2(x-1)(x-4)$

Question 2 of 7

Find the equation of the graph below

1. $y = - (x + 3) (x - 5)$ $y=-(x+3)(x-5)$
2. $y = (x - 3) (x - 5)$ $y=(x-3)(x-5)$
3. $y = (x + 3) (x - 5)$ $y=(x+3)(x-5)$
4. $y = - (x + 3) (x + 5)$ $y=-(x+3)(x+5)$

Question 3 of 7

Find the equation of the parabola given the following information

x = - 1

$x=-1$

x = 5

$x=5$
Vertex:

(2, - 9)

$(2,-9)$

1. $y = x^{2} - 4 x - 5$ $y=x^2-4x-5$
2. $y = x^{2} + 4 x - 5$ $y=x^2+4x-5$
3. $y = x^{2} - 4 x - 6$ $y=x^2-4x-6$
4. $y = x^{2} - 2 x - 5$ $y=x^2-2x-5$

Question 4 of 7

Find the equation of the parabola given the following information

x = - 2

$x=-2$

x = 4

$x=4$
Vertex:

(2, - 8)

$(2,-8)$

1. $y = x^{2} + 2 x + 8$ $y=x^2+2x+8$
2. $y = x^{2} - 2 x + 8$ $y=x^2-2x+8$
3. $y = x^{2} - 2 x - 8$ $y=x^2-2x-8$
4. $y = x^{2} - 8 x + 2$ $y=x^2-8x+2$

Question 5 of 7

Find the equation of the parabola given the following information

x = - 2

$x=-2$

x = 3

$x=3$
Vertex:

(2, 8)

$(2,8)$

1. $y = x^{2} - x - 6$ $y=x^2-x-6$
2. $y = - 2 x^{2} + 2 x + 12$ $y=-2x^2+2x+12$
3. $y = 2 x^{2} - 2 x - 12$ $y=2x^2-2x-12$
4. $y = 2 x^{2} + 2 x + 6$ $y=2x^2+2x+6$

Question 6 of 7

Find the equation of the parabola if

$(0,8)$ and

$(2,0)$ are points on the line.

1. $y=(x-2)^2$
2. $y=(x+2)^2$
3. $y=(x-4)^2$
4. $y=(x+4)^2$

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Equation of a Parabola

$y=\color{#ff0090}{a}\color{green}{x}^{2}+\color{blue}{b}\color{green}{x}+c$

$\color{green}{x}=-\frac{\color{blue}{b}}{2\color{#ff0090}{a}}$

First, solve for $b$ using the value of the $x$ -intercept.

$x$	$=$	$\frac{-\color{blue}{b}}{2\color{#ff0090}{a}}$

$2$	$=$	$\frac{-\color{blue}{b}}{2\color{#ff0090}{a}}$	$x=2$

$4$ $a$	$=$	$-b$	Multiply both sides by $2a$
$-4$ $a$	$=$	$b$	Divide both sides by $-1$
$b$	$=$	$-4$ $a$

Substitute the values of $b$ and

$c$ into the general formula.

$y$	$=$	$\color{#ff0090}{a}\color{green}{x}^{2}+\color{blue}{b}\color{green}{x}+c$
$y$	$=$	$\color{#ff0090}{a}\color{green}{x}^{2}+\color{blue}{-4a}\color{green}{x}+8$	$b=-4a$ , $c=8$

Substitute the value of $x$ and

$y$ into the resulting equation using the

$x$ -intercept of the graph to solve for $a$ .

$y$	$=$	$\color{#ff0090}{a}\color{green}{x}^{2}+(-4\color{#ff0090}{a})\color{green}{x}+8$
$0$	$=$	$\color{#ff0090}{a}\color{green}{2}^{2}+(-4\color{#ff0090}{a})(\color{green}{2})+8$	$x=2$ , $y=0$
$0$	$=$	$4$ $a$ $-8$ $a$ $+8$	Simplify
$0$	$=$	$-4$ $a$ $+8$
$-8$	$=$	$-4$ $a$	Subtract $8$ from both sides
$2$	$=$	$a$	Divide both sides by $2$
$a$	$=$	$2$	Divide both sides by $2$

Substitute the value of $a$ into the equation for $b$ .

$b$	$=$	$-4$ $a$
$b$	$=$	$-4$ $(2)$	$a=2$
$b$	$=$	$-8$

Substitute the value of $a$ , $b$ and

$c$ into the general formula.

$y$	$=$	$a$ $x^2+$ $b$ $x+c$
$y$	$=$	$2$ $x^2+$ $(-8)$ $x+8$	$a=2$ , $b=-8$ , $c=8$
$y$	$=$	$2x^2-8x+8$
$y$	$=$	$x^2-4x+4$	Divide the expression by $2$
$y$	$=$	$(x-2)^2$	Express as a square of binomial

$y=(x-2)^2$

Question 7 of 7

Find the equation of the parabola if

$(1,8)$ and

$(0,6)$ are points on the line.

1. $y=(x-3)(x+2)$
2. $y=(x+3)(x+1)$
3. $y=2(x-3)(x-1)$
4. $y=-2(x-3)(x+1)$

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Equation of a Parabola

$y=\color{#ff0090}{a}\color{green}{x}^{2}+\color{blue}{b}\color{green}{x}+c$

$\color{green}{x}=-\frac{\color{blue}{b}}{2\color{#ff0090}{a}}$

First, solve for $b$ using the value of the $x$ in the point

$(1,8)$ .

$x$	$=$	$\frac{-\color{blue}{b}}{2\color{#ff0090}{a}}$

$1$	$=$	$\frac{-\color{blue}{b}}{2\color{#ff0090}{a}}$	$x=1$

$-2$ $a$	$=$	$b$	Multiply both sides by $-2a$
$b$	$=$	$-2$ $a$

Substitute the values of $b$ and

$c$ into the general formula.

$y$	$=$	$\color{#ff0090}{a}\color{green}{x}^{2}+\color{blue}{b}\color{green}{x}+c$
$y$	$=$	$\color{#ff0090}{a}\color{green}{x}^{2}+\color{blue}{-2a}\color{green}{x}+6$	$b=-2a$ , $c=6$

Substitute the value of $x$ and

$y$ into the resulting equation using the point

$(1,8)$ to solve for $a$ .

$y$	$=$	$\color{#ff0090}{a}\color{green}{x}^{2}+(-2\color{#ff0090}{a})\color{green}{x}+6$
$8$	$=$	$\color{#ff0090}{a}\color{green}{1}^{2}+(-2\color{#ff0090}{a})(\color{green}{2})+6$	$x=1$ , $y=8$
$8$	$=$	$a$ $-2$ $a$ $+6$	Simplify
$8$	$=$	$-a$ $+6$
$2$	$=$	$-a$	Subtract $6$ from both sides
$-2$	$=$	$a$	Divide both sides by $-1$
$a$	$=$	$-2$	Divide both sides by $2$

Substitute the value of $a$ into the equation for $b$ .

$b$	$=$	$-2$ $a$
$b$	$=$	$-2$ $(-2)$	$a=-2$
$b$	$=$	$4$

Substitute the value of $a$ , $b$ and

$c$ into the general formula.

$y$	$=$	$a$ $x^2+$ $b$ $x+c$
$y$	$=$	$-2$ $x^2+$ $4$ $x+6$	$a=-2$ , $b=4$ , $c=6$
$y$	$=$	$-2x^2+4x+6$
$y$	$=$	$-2(x^2-2x-3)$	Factor out $2$

Since the equation is in standard form

$($ $a$

$x^2+$ $b$

$x+$ $c$

$=0)$ we can factorise using the cross method.

$x^2$ $-2$

$x$ $-3$

$=0$

To factorise, we need to find two numbers that add to $-2$ and multiply to $-3$

$-3$ and

$1$ fit both conditions

$-3 + 1$	$=$	$-2$
$-3 xx 1$	$=$	$-3$

Read across to get the factors.

$(x-3)(x+1)$

$y=-2(x-3)(x+1)$

$p$ $p$	$=$ $=$	$1$ $1$	$x$ $x$ -intercept
$q$ $q$	$=$ $=$	$4$ $4$	$x$ $x$ -intercept
$y$ $y$	$=$ $=$	$4$ $4$	$y$ $y$ -intercept
$x$ $x$	$=$ $=$	$0$ $0$	Value of $x$ $x$ at the $y$ $y$ -intercept

$y$ $y$	$=$ $=$	$a (x -$ $a(x-$ $p$ $p$ $) (x -$ $)(x-$ $q$ $q$ $)$ $)$	Intercept Form
$4$ $4$	$=$ $=$	$a (0 -$ $a(0-$ $1$ $1$ $) (0 -$ $)(0-$ $4$ $4$ $)$ $)$	Substitute values
$4$ $4$	$=$ $=$	$a (- 1) (- 4)$ $a(-1)(-4)$
$4$ $4$	$=$ $=$	$4 a$ $4a$
$4$ $4$ $\div 4$ $-:4$	$=$ $=$	$4 a$ $4a$ $\div 4$ $-:4$
$1$ $1$	$=$ $=$	$a$ $a$
$a$ $a$	$=$ $=$	$1$ $1$

$y$ $y$	$=$ $=$	$a (x -$ $a(x-$ $p$ $p$ $) (x -$ $)(x-$ $q$ $q$ $)$ $)$	Intercept Form
$y$ $y$	$=$ $=$	$1 (x -$ $1(x-$ $1$ $1$ $) (x -$ $)(x-$ $4$ $4$ $)$ $)$	Substitute values
$y$ $y$	$=$ $=$	$(x - 1) (x - 4)$ $(x-1)(x-4)$

$p$ $p$	$=$ $=$	$- 3$ $-3$	$x$ $x$ -intercept
$q$ $q$	$=$ $=$	$5$ $5$	$x$ $x$ -intercept
$y$ $y$	$=$ $=$	$15$ $15$	$y$ $y$ -intercept
$x$ $x$	$=$ $=$	$0$ $0$	Value of $x$ $x$ at the $y$ $y$ -intercept

$y$ $y$	$=$ $=$	$a (x -$ $a(x-$ $p$ $p$ $) (x -$ $)(x-$ $q$ $q$ $)$ $)$	Intercept Form
$15$ $15$	$=$ $=$	$a (0 - ($ $a(0-($ $- 3$ $-3$ $)) (0 -$ $))(0-$ $5$ $5$ $)$ $)$	Substitute values
$15$ $15$	$=$ $=$	$a (3) (- 5)$ $a(3)(-5)$
$15$ $15$	$=$ $=$	$- 15 a$ $-15a$
$15$ $15$ $\div (- 15)$ $-:(-15)$	$=$ $=$	$- 15 a$ $-15a$ $\div (- 15)$ $-:(-15)$
$- 1$ $-1$	$=$ $=$	$a$ $a$
$a$ $a$	$=$ $=$	$- 1$ $-1$

Write a Parabola from the Graph

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Quiz summary

Information

1. Question

Hint

Great Work!

Intercept Form

2. Question

Hint

Well Done!

Intercept Form

3. Question

Hint

Excellent!

Equation of a Parabola

4. Question

Hint

Nice Job!

Equation of a Parabola

5. Question

Hint

Well Done!

Equation of a Parabola

6. Question

Hint

Great Work!

Equation of a Parabola

7. Question

Hint

Fantastic!

Equation of a Parabola

Quizzes

$y$ $y$	$=$ $=$	$a$ $a$ $(x + 1) (x - 5)$ $(x+1)(x-5)$
$- 9$ $-9$	$=$ $=$	$a$ $a$ $($ $($ $2$ $2$ $+ 1) ($ $+1)($ $2$ $2$ $- 5)$ $-5)$	$x = 2$ $x=2$ and $y = - 9$ $y=-9$
$- 9$ $-9$	$=$ $=$	$a$ $a$ $(3) (- 3)$ $(3)(-3)$	Simplify
$- 9$ $-9$	$=$ $=$	$- 9$ $-9$ $a$ $a$
$1$ $1$	$=$ $=$	$a$ $a$	Divide both sides by $- 9$ $-9$
$a$ $a$	$=$ $=$	$1$ $1$

$y$ $y$	$=$ $=$	$a$ $a$ $(x + 2) (x - 4)$ $(x+2)(x-4)$
$- 8$ $-8$	$=$ $=$	$a$ $a$ $($ $($ $2$ $2$ $+ 2) ($ $+2)($ $2$ $2$ $- 4)$ $-4)$	$x = 2$ $x=2$ and $y = - 8$ $y=-8$
$- 8$ $-8$	$=$ $=$	$a$ $a$ $(4) (- 2)$ $(4)(-2)$	Simplify
$- 8$ $-8$	$=$ $=$	$- 8$ $-8$ $a$ $a$
$1$ $1$	$=$ $=$	$a$ $a$	Divide both sides by $- 8$ $-8$
$a$ $a$	$=$ $=$	$1$ $1$

$y$ $y$	$=$ $=$	$a$ $a$ $(x + 2) (x - 3)$ $(x+2)(x-3)$
$8$ $8$	$=$ $=$	$a$ $a$ $($ $($ $2$ $2$ $+ 2) ($ $+2)($ $2$ $2$ $- 3)$ $-3)$	$x = 2$ $x=2$ and $y = 8$ $y=8$
$8$ $8$	$=$ $=$	$a$ $a$ $(4) (- 1)$ $(4)(-1)$	Simplify
$8$ $8$	$=$ $=$	$- 4$ $-4$ $a$ $a$
$- 2$ $-2$	$=$ $=$	$a$ $a$	Divide both sides by $- 4$ $-4$
$a$ $a$	$=$	$-2$

$y$	$=$	$a$ $(x+2)(x-3)$
$y$	$=$	$-2$ $(x+2)(x-3)$	$a=-2$
$y$	$=$	$-2(x^2-x-6)$
$y$	$=$	$-2x^2+2x+12$	Simplify the equation