3.9: Vertex Form of a Quadratic Function
Given the equation , list the transformations of
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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 is . You can apply transformations to the graph of to create a new graph with a corresponding new equation. This new equation can be written in vertex form. The vertex form of a quadratic function is where:
- is the vertical stretch factor. If is negative, there is a vertical reflection and the parabola will open downwards.
- is the vertical translation.
- 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:
- – Is negative? YES. The parabola will open downwards.
- – 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 is .
- – 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 .
- – Is there a number after the variable ‘ ’? 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 in vertex form.
- Vertical stretch by a factor of 3
- Vertical translation up 5 units
- Horizontal translation left 4 units
Solution:
- – The image is not reflected in the -axis. A negative sign is not required.
- – The vertical stretch is 3, so .
- – The vertical translation is 5 units up, so .
- – The horizontal translation is 4 units left so .
The equation of the image of is .
Example C
Using as the base function, identify the transformations that have occurred to produce the following image graph. Use these transformations to write the equation in vertex form.
Solution:
– The parabola does not open downward so will be positive.
– The -values of 1 and 4 are now up 3 and up 12. .
– The coordinate of the vertex is –5 so .
– The coordinate of the vertex is +3 so .
The equation is .
Example D
In general, the mapping rule used to generate the image of a function is where are the coordinates of the image graph. The resulting mapping rule from to the image is . The mapping rule details the transformations that were applied to the coordinates of the base function .
Given the following quadratic equation, 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.
These new coordinates of the image graph can be plotted to generate the graph.
Concept Problem Revisited
Given the equation , list the transformations of .
so the vertical stretch is 3 . so the vertical translation is up 2 . so the horizontal translation is left 4 .
Vocabulary
- Horizontal translation
- The horizontal translation is the change in the base graph that shifts the graph right or left. It changes the coordinate of the vertex.
- Mapping Rule
- The mapping rule defines the transformations that have occurred to a function. The mapping rule is where are the coordinates of the image graph.
- Transformation
- A transformation is any change in the base graph . The transformations that apply to the parabola are a horizontal translation, a vertical translation, a vertical stretch and a vertical reflection.
- Vertex form of
- The vertex form of is the form of the quadratic base function that shows the transformations of the image graph. The vertex form of the equation is .
- Vertical Reflection
- The vertical reflection is the reflection of the image graph in the -axis. The graph opens downward and the -values are negative values.
- Vertical Stretch
- The vertical stretch is the change made to the base function 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 .
- Vertical Translation
- The vertical translation is the change in the base graph that shifts the graph up or down. It changes the coordinate of the vertex.
Guided Practice
1. Identify the transformations of for the quadratic function
2. List the transformations of and graph the function
3. Graph the function using the mapping rule method.
Answers:
1. Rewrite the equation in vertex form. – is negative so the parabola opens downwards.
– The vertical stretch of this function is .
– The vertical translation is -3 .
– The horizontal translation is +4 .
2.
3. Mapping Rule
Make a table of values:
–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 in each of the given functions:
Graph the following quadratic functions.
Using the following mapping rules, write the equation, in vertex form, that represents the image of .
Image Attributions
Description
Learning Objectives
Here you will learn to write the equation for a parabola that has undergone transformations.