Activity
1. Use the graphing tool or technology outside the course. Graph , and then experiment with adding different linear terms (for example, , , ). Record your observations.
Compare your answer:
The graph still looks like a quadratic function when adding a linear term but it "shifts" both horizontally and vertically. Adding a positive multiple of to shifts it down and to the left while adding a negative multiple of shifts it down and to the right.
Use your observations from question 1 to answer questions 2 – 5 and complete the table.
equation | -intercepts | -coordinate of vertex |
|
2a. _____ |
2b. _____ |
|
3a. _____ |
3b. _____ |
|
4a. _____ |
4b. _____ |
|
5a. _____ |
5b. _____ |
2. Find the -intercepts and the -coordinate of the vertex for the equation .
a. What are the -intercepts for the equation ?
Compare your answer:
,
b. What is the -coordinate of the vertex for the equation ?
-3
3. Find the -intercepts and the -coordinate of the vertex for the equation .
a. What are the -intercepts for the equation ?
Compare your answer:
,
b. What is the -coordinate of the vertex for the equation ?
5
4. Find the -intercepts and the -coordinate of the vertex for the equation .
a. What are the -intercepts for the equation ?
Compare your answer:
,
b. What is the -coordinate of the vertex for the equation ?
25
5. Find the -intercepts and the -coordinate of the vertex for the equation .
a. Find the -intercepts for the equation .
Compare your answer:
,
b. What is the -coordinate of the vertex for the equation ?
-18
Some quadratic expressions have no linear terms. For questions 6 – 7, find the -intercepts and the -coordinate of the vertex of the graph representing each equation:
(Note it is possible for the graph to not intersect the -axis.) If you get stuck, try graphing the equations.
6. Find the -intercepts and the -coordinate of the vertex for the equation .
a. What are the -intercepts for the equation ?
Compare your answer:
,
b. What is the -coordinate of the vertex for the equation ?
0
7. Find the -intercepts and the -coordinate of the vertex for the equation .
a. What are the -intercepts for the equation ?
Compare your answer:
There are no -intercepts.
b. What is the -coordinate of the vertex for the equation ?
0
Self Check
Additional Resources
Quadratics Without Linear Terms
Example
Find the -intercepts and the vertex for .
Step 1 - Write the function in factored form. Use difference of squares:
Step 2 - Set each factor equal to 0.
The -intercepts are and .
Step 3 - Find the -value in the middle of the two -intercepts. This occurs at . This is the -coordinate of the vertex.
To find the point at the vertex, substitute into the function:
Vertex:
Try it
Try It: Quadratics Without Linear Terms
Find the -intercepts and the vertex for .
Here is how to find the -intercepts and vertex of :
Step 1 - Write the function in factored form:
Step 2 - Set each factor equal to 0.
The -intercepts are at and .
The vertex is located halfway between the -intercepts, at .
Step 3 - To find the point, substitute into .
The vertex is at .