Activity
- Consider the function defined by .
Then, answer questions a–g, and create a table of values in which to enter the given x values and their outputs using the function.
a. When the value of , what is the value of ?
0
b. When the value of , what is the value of ?
-3
c. When the value of , what is the value of ?
0
d. When the value of , what is the value of ?
5
e. When the value of , what is the value of ?
12
f. When the value of , what is the value of ?
21
g. When the value of , what is the value of ?
45
2. Now, answer questions a–d to analyze the function defined by .
a. What are the -intercepts of the graph? Be prepared to show your reasoning.
Compare your answer:
The -intercepts for function are and . These points have an -coordinate that makes the output value 0.
b. What is the value of the -coordinate in the vertex of the graph?
-2
c. What is the value of the -coordinate in the vertex of the graph?
-4
d. What is the vertex of the graph of function ? Be prepared to show your reasoning.
Compare your answer:
The vertex appears to be at . From the tables, the function values seem to change from decreasing to increasing at those points.
3. Consider the function defined by . Then, answer questions a–g, and create a table of values in which to enter the given x values and their outputs using the function.
a. When the value of , what is the value of ?
32
b. When the value of , what is the value of ?
21
c. When the value of , what is the value of ?
0
d. When the value of , what is the value of ?
-3
e. When the value of , what is the value of ?
-4
f. When the value of , what is the value of ?
0
g. When the value of , what is the value of ?
5
4. Now, answer questions a–d to analyze the function defined by .
a. What are the -intercepts of the graph? Be prepared to show your reasoning.
Compare your answer:
The -intercepts for function are and . These points have an -coordinate that makes the output value 0.
b. What is the value of the -coordinate in the vertex of the graph?
2
c. What is the value of the -coordinate in the vertex of the graph?
-4
d. What is the vertex of the graph of function ? Be prepared to show your reasoning.
Compare your answer:
The vertex appears to be at . From the tables, the function values seem to change from decreasing to increasing at those points.
5. Use the graphing tool or technology outside the course.
a. Plot the points from the tables on the same coordinate plane. (Consider using different colors or markings for each set of points so you can tell them apart.)
Compare your answer:
b. Make a couple of observations about how the two graphs compare.
Compare your answer:
- The -intercept of both graphs is at .
- The graphs seem to be a reflection of each other across the -axis.
- The vertices both have a -coordinate of -4 and also mirror one another across the -axis.
- Each function mirrors itself across a vertical line through its vertex.
Self Check
Additional Resources
Determining Parts of a Graph from a Table
Identify the -intercepts and vertex of the quadratic from the table:
-5 |
-3 |
-1 |
0 |
1 |
|
16 |
0 |
-2 |
0 |
16 |
The -intercepts are where the -values are 0.
This occurs at and .
The vertex occurs where the function will create a mirror image, or be symmetrical for all points. This often occurs halfway between the two zeros (when there are two zeros).
For this function, the vertex is at .
Try it
Try It: Determining Parts of a Graph from a Table
Determine the vertex from the table:
5 |
6 |
7 |
8 |
9 |
10 |
|
0 |
4 |
8 |
4 |
0 |
-2 |
Here is how to determine the vertex from the table:
The zeros are located at and . The vertex will be halfway between the two zeros. This occurs at .
The point in the table at is .
This is the vertex.