Write a Quadratic Equation Given the Vertex and Another Point

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Question 1 of 2

Find the equation of the graph below

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

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Vertex Form

$y=a{(x-\color{#007DDC}{h})}^2+\color{#e65021}{k}$

where

$($ $h$

$,$ $k$

$)$ is the vertex

Since the graph indicates the vertex, use the Vertex Form. Slot

$($ $h$

$,$ $k$

$)$ and

$(x,y)$ into the Vertex Form to solve for

$a$ . Then, substitute

$a$ , $h$ and $k$ back to the main formula to form an equation.

First, label values from the graph

Note that the vertex is at

$(-2,3)$

$h$	$=$	$-2$	from vertex
$k$	$=$	$3$	from vertex
$x$	$=$	$-5$	$x$ intercept
$y$	$=$	$0$	value of $y$ at $x$ intercept

Now, slot these values into the Vertex Form and solve for

$a$

$y$	$=$	$a{(\color{#00880A}{x}-\color{#007DDC}{h})}^2+\color{#e65021}{k}$	Vertex Form
$0$	$=$	$a{(\color{#00880A}{-5}-(\color{#007DDC}{-2}))}^2+\color{#e65021}{3}$	Substitute values
$0$	$=$	$a(-3)^2+3$
$0$	$=$	$9a+3$
$9a+3$	$=$	$0$
$9a+3$ $-3$	$=$	$0$ $-3$	Subtract $3$ from both sides
$9a$ $divide9$	$=$	$-3$ $divide9$	Divide both sides by $9$

$a$	$=$	$-3/9$

$a$	$=$	$-1/3$	Simplify

Finally, substitute

$a$ ,

$h$ and

$k$ into the Vertex Form

$y$	$=$	$a{(x-\color{#007DDC}{h})}^2+\color{#e65021}{k}$	Vertex Form

$y$	$=$	$-\frac{1}{3}{(x-(\color{#007DDC}{-2}))}^2+\color{#e65021}{3}$	Substitute values

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

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

Question 2 of 2

Find the equation of the graph below

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

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Vertex Form

$y=a{(x-\color{#007DDC}{h})}^2+\color{#e65021}{k}$

where

$($ $h$

$,$ $k$

$)$ is the vertex

Since the graph indicates the vertex, use the Vertex Form. Slot

$($ $h$

$,$ $k$

$)$ and

$(x,y)$ into the Vertex Form to solve for

$a$ . Then, substitute

$a$ , $h$ and $k$ back to the main formula to form an equation.

First, label values from the graph

$h$	$=$	$3$	from vertex
$k$	$=$	$2$	from vertex
$x$	$=$	$4$	from given point
$y$	$=$	$1$	from given point

Now, slot these values into the Vertex Form and solve for

$a$

$y$	$=$	$a{(\color{#00880A}{x}-\color{#007DDC}{h})}^2+\color{#e65021}{k}$	Vertex Form
$1$	$=$	$a{(\color{#00880A}{4}-\color{#007DDC}{3})}^2+\color{#e65021}{2}$	Substitute values
$1$	$=$	$a(1)^2+2$
$1$	$=$	$a+2$
$a+2$	$=$	$1$
$a+2$ $-2$	$=$	$1$ $-2$	Subtract $2$ from both sides
$a$	$=$	$-1$

Finally, substitute

$a$ ,

$h$ and

$k$ into the Vertex Form

$y$	$=$	$a{(x-\color{#007DDC}{h})}^2+\color{#e65021}{k}$	Vertex Form
$y$	$=$	$-1{(x-\color{#007DDC}{3})}^2+\color{#e65021}{2}$	Substitute values
$y$	$=$	$-(x-3)^2+2$

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

Write a Quadratic Equation Given the Vertex and Another Point

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Quiz summary

Information

1. Question

Hint

Correct!

Vertex Form

2. Question

Hint

Excellent!

Vertex Form

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