3.9: Vertex Form of a Quadratic Function
Given the equation
Watch This
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
When the equation of the basic quadratic function is written in vertex form, the function can also be expressed in mapping notation form. This form describes how to obtain the image of a given graph by using the changes in the ordered pairs.
The standard base table of values for the base quadratic function
When these ordered pairs are plotted, we get the base parabola. The mapping rule used to generate the image of a quadratic function is
Given the following quadratic equation, \begin{align*}y=2(x+3)^2+5\end{align*} write the mapping rule and create a table of values for the mapping rule.
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*}, list the transformations of \begin{align*}y=x^2\end{align*}.
\begin{align*}a=3\end{align*} so the vertical stretch is 3. \begin{align*}k=2\end{align*} so the vertical translation is up 2. \begin{align*}h=4\end{align*} so the horizontal translation is left 4.
Vocabulary
 Horizontal translation
 The horizontal translation is the change in the base graph \begin{align*}y=x^2\end{align*} that shifts the graph right or left. It changes the \begin{align*}x\end{align*}coordinate of the vertex.
 Mapping Rule
 The mapping rule is another form used to express a quadratic function. The mapping rule defines the transformations that have occurred to the base quadratic function \begin{align*}y=x^2\end{align*}. The mapping rule is \begin{align*}(x,y) \rightarrow (x^\prime,y^\prime)\end{align*} where \begin{align*}(x^\prime,y^\prime)\end{align*} are the coordinates of the image graph.
 Transformation
 A transformation is any change in the base graph \begin{align*}y=x^2\end{align*}. 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*}
 The vertex form of \begin{align*}y = x^2\end{align*} is the form of the quadratic base function \begin{align*}y=x^2\end{align*} that shows the transformations of the image graph. The vertex form of the equation is \begin{align*} y=a(xh)^2+k\end{align*}.
 Vertical Reflection
 The vertical reflection is the reflection of the image graph in the \begin{align*}x\end{align*}axis. The graph opens downward and the \begin{align*}y\end{align*}values are negative values.
 Vertical Stretch
 The vertical stretch is the change made to the base function \begin{align*}y=x^2\end{align*} 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*}.
 Vertical Translation
 The vertical translation is the change in the base graph \begin{align*}y=x^2\end{align*} that shifts the graph up or down. It changes the \begin{align*}y\end{align*}coordinate of the vertex.
Guided Practice
1. Identify the transformations of \begin{align*}y=x^2\end{align*} for the quadratic function \begin{align*}2(y+3)=(x4)^2\end{align*}
2. List the transformations of \begin{align*}y=x^2\end{align*} and graph the function \begin{align*}=(x+5)^2+4\end{align*}
3. Graph the function \begin{align*}y=2(x2)^2+3\end{align*} using the mapping rule method.
Answers:
1. \begin{align*}a\end{align*} – \begin{align*}a\end{align*} is negative so the parabola opens downwards.
\begin{align*}a\end{align*} – The vertical stretch of this function is \begin{align*}\frac{1}{2}\end{align*}.
\begin{align*}k\end{align*} – The vertical translation is 3.
\begin{align*}h\end{align*} – The horizontal translation is +4.
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 x+2\end{align*}  \begin{align*}y \rightarrow 2y+3\end{align*}  

\begin{align*}3\end{align*}  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
Complete the following table to identify the transformations of \begin{align*}y=x^2\end{align*} in each of the given functions:
Number  \begin{align*}a\end{align*}  \begin{align*}k\end{align*}  \begin{align*}h\end{align*} 

1.  
2.  
3.  
4.  
5. 
 \begin{align*}y=4(x2)^29\end{align*}
 \begin{align*}y=\frac{1}{6}x^2+7\end{align*}
 \begin{align*}=3(x1)^26\end{align*}
 \begin{align*}y=\frac{1}{5}(x+4)^2+3\end{align*}
 \begin{align*}y=5(x+2)^2\end{align*}
Graph the following quadratic functions using the mapping rule method:
 \begin{align*}y=2(x4)^25\end{align*}
 \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*}
Image Attributions
Description
Learning Objectives
Here you will learn to write the equation for a parabola that has undergone transformations.
Difficulty Level:
At GradeTags:
Subjects:
Date Created:
Dec 19, 2012Last Modified:
Apr 29, 2014Vocabulary
If you would like to associate files with this Modality, please make a copy first.