3.9: Vertex Form of a Quadratic Function
Given the equation
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James Sousa: Find the Equation of a Quadratic Function from a Graph
Guidance
The equation for a basic parabola with a vertex at

a is the vertical stretch factor. Ifa is negative, there is a vertical reflection and the parabola will open downwards. 
k is the vertical translation. 
h is the horizontal translation.
Given the equation of a parabola in vertex form, you should be able to sketch its graph by performing transformations on the basic parabola. This process is shown in the examples.
Example A
Given the following function in vertex form, identify the transformations of
Solution:

a – Isa negative? YES. The parabola will open downwards.

a – Is there a number in front of the squared portion of the equation? YES. The vertical stretch factor is the absolute value of this number. Therefore, the vertical stretch of this function is12 .

k – Is there a number after the squared portion of the equation? YES. The value of this number is the vertical translation. The vertical translation is –1.

h – Is there a number after the variable ‘x ’? YES. The value of this number is the opposite of the sign that appears in the equation. The horizontal translation is +2.
Example B
Given the following transformations, determine the equation of the image of
 Vertical stretch by a factor of 3
 Vertical translation up 5 units
 Horizontal translation left 4 units
Solution:

a – The image is not reflected in thex axis. A negative sign is not required.

a – The vertical stretch is 3, soa=3 .

k – The vertical translation is 5 units up, sok=5 .

h – The horizontal translation is 4 units left soh=−4 .
The equation of the image of
Example C
Using
Solution:
The equation is
Example D
In general, the mapping rule used to generate the image of a function is
Given the following quadratic equation,
Solution:
The mapping rule for this function will tell exactly what changes were applied to the coordinates of the base quadratic function.
\begin{align*}y=2(x+3)^2+5: \quad (x,y) \rightarrow (x3,2y+5)\end{align*}
These new coordinates of the image graph can be plotted to generate the graph.
Concept Problem Revisited
Given the equation \begin{align*}y=3(x+4)^2+2\end{align*}
\begin{align*}a=3\end{align*}
Vocabulary
 Horizontal translation

The horizontal translation is the change in the base graph \begin{align*}y=x^2\end{align*}
y=x2 that shifts the graph right or left. It changes the \begin{align*}x\end{align*}x− coordinate of the vertex.
 Mapping Rule

The mapping rule defines the transformations that have occurred to a function. The mapping rule is \begin{align*}(x,y) \rightarrow (x^\prime,y^\prime)\end{align*}
(x,y)→(x′,y′) where \begin{align*}(x^\prime,y^\prime)\end{align*}(x′,y′) are the coordinates of the image graph.
 Transformation

A transformation is any change in the base graph \begin{align*}y=x^2\end{align*}
y=x2 . The transformations that apply to the parabola are a horizontal translation, a vertical translation, a vertical stretch and a vertical reflection.

Vertex form of \begin{align*}y = x^2\end{align*}
y=x2 
The vertex form of \begin{align*}y = x^2\end{align*}
y=x2 is the form of the quadratic base function \begin{align*}y=x^2\end{align*}y=x2 that shows the transformations of the image graph. The vertex form of the equation is \begin{align*} y=a(xh)^2+k\end{align*}y=a(x−h)2+k .
 Vertical Reflection

The vertical reflection is the reflection of the image graph in the \begin{align*}x\end{align*}
x axis. The graph opens downward and the \begin{align*}y\end{align*}y values are negative values.
 Vertical Stretch

The vertical stretch is the change made to the base function \begin{align*}y=x^2\end{align*}
y=x2 by stretching (or compressing) the graph vertically. The vertical stretch will produce an image graph that appears narrower (or wider) then the original base graph of \begin{align*}y=x^2\end{align*}y=x2 .
 Vertical Translation

The vertical translation is the change in the base graph \begin{align*}y=x^2\end{align*}
y=x2 that shifts the graph up or down. It changes the \begin{align*}y\end{align*}y− coordinate of the vertex.
Guided Practice
1. Identify the transformations of \begin{align*}y=x^2\end{align*}
2. List the transformations of \begin{align*}y=x^2\end{align*}
3. Graph the function \begin{align*}y=2(x2)^2+3\end{align*}
Answers:
1. Rewrite the equation in vertex form. \begin{align*}a\end{align*}
\begin{align*}a\end{align*}
\begin{align*}k\end{align*}
\begin{align*}h\end{align*}
2.
\begin{align*}a & \rightarrow negative\\ a & \rightarrow 1\\ k & \rightarrow +4\\ h & \rightarrow 5\end{align*}
3. Mapping Rule \begin{align*}(x,y) \rightarrow (x+2,2y+3)\end{align*}
Make a table of values:
\begin{align*}x \rightarrow\end{align*} 
\begin{align*}x+2\end{align*} 
\begin{align*}y \rightarrow\end{align*} 
\begin{align*}2y+3\end{align*} 


–3  –1  9  21  
–2  0  4  11  
–1  1  1  5  
0  2  0  3  
1  3  1  5  
2  4  4  11  
3  5  9  21 
Draw the graph:
Practice
Identify the transformations of \begin{align*}y=x^2\end{align*}

\begin{align*}y=4(x2)^29\end{align*}
y=4(x−2)2−9 
\begin{align*}y=\frac{1}{6}x^2+7\end{align*}
y=−16x2+7 
\begin{align*}y=3(x1)^26\end{align*}
y=−3(x−1)2−6 
\begin{align*}y=\frac{1}{5}(x+4)^2+3\end{align*}
y=15(x+4)2+3 
\begin{align*}y=5(x+2)^2\end{align*}
y=5(x+2)2
Graph the following quadratic functions.

\begin{align*}y=2(x4)^25\end{align*}
y=2(x−4)2−5  \begin{align*}y=\frac{1}{3}(x2)^2+6\end{align*}
 \begin{align*}y=2(x+3)^2+7\end{align*}
 \begin{align*}y=\frac{1}{2}(x+6)^2+9\end{align*}
 \begin{align*}y=\frac{1}{3}(x4)^2\end{align*}
Using the following mapping rules, write the equation, in vertex form, that represents the image of \begin{align*}y = x^2\end{align*}.
 \begin{align*}(x,y) \rightarrow \left(x+1, \frac{1}{2}y\right)\end{align*}
 \begin{align*}(x,y) \rightarrow (x+6,2y3)\end{align*}
 \begin{align*}(x,y) \rightarrow \left(x1, \frac{2}{3}y+2\right)\end{align*}
 \begin{align*}(x,y) \rightarrow (x+3,3y+1)\end{align*}
 \begin{align*}(x,y) \rightarrow \left(x5,\frac{1}{3}y7\right)\end{align*}
Absolute Value
The absolute value of a number is the distance the number is from zero. Absolute values are never negative.quadratic function
A quadratic function is a function that can be written in the form , where , , and are real constants and .Vertex form
The vertex form of a parabola is or where is the vertex.Image Attributions
Description
Learning Objectives
Here you will learn to write the equation for a parabola that has undergone transformations.