Activity
1. Here is one way to rewrite in vertex form (of a quadratic expression).
a. Study the steps and write a brief explanation of what is happening at each step.
Original expression
Step 1 -
Step 2 -
Step 3 -
Step 4 -
Step 5 -
Compare your answers:
Original expression
Step 1 - Rewrite the expression with 3 as a factor.
Step 2 - Add 4 to complete the square. Subtract 4 to keep the value of the expression the same.
Step 3 - Combine like terms so we can see the perfect square, .
Step 4 - Rewrite as a squared expression.
Step 5 - Apply the distributive property to put the expression in vertex form.
b. What is the -coordinate of the vertex (of a graph) that represents this expression?
The -coordinate of the vertex is -2.
c. What is the -coordinate of the vertex of the graph that represents this expression?
The -coordinate of the vertex is -3.
d. Does the graph open upward or downward? Explain how you know.
Compare your answer:
The graph opens upward. For example, in an earlier unit, we saw that when the coefficient of the squared term is positive, the graph opens upward.
2. Rewrite in vertex form.
Compare your answer:
3. Rewrite in vertex form.
Compare your answer:
4. Rewrite in vertex form.
Compare your answer:
Are you ready for more?
Extending Your Thinking
1. Fill in the missing parts of the function in vertex form without completing the square. (Hint: Think about finding the zeros of the function.)
a. What is the value of a.
2
b. What is the sign of b.
Multiple Choice:
Subtraction(-)
Addition(+)
Minus (-)
c. What is the value of c.
6
d. What is the sign of d.
Multiple Choice:
Subtraction(-)
Addition(+)
Minus (-)
e. What is the value of e.
18
f. Be prepared to show your reasoning for your answers.
Compare your answer:
The zeros of are 3 and 9. Because the graph of a quadratic function has a vertical line of symmetry across the vertex, the vertex of the graph must have -coordinate 6. The -coordinate of the vertex is -18 because . This means can be expressed as . If we expand , we can see that the coefficient of is 2. This means that in the vertex form has to be 2 as well.
2. Fill in the missing parts of the function in vertex form without completing the square.
a. What is the value of a.
2
b. What is the sign of b.
Multiple Choice:
Subtraction(-)
Addition(+)
Minus (-)
c. What is the value of c.
6
d. What is the sign of d.
Multiple Choice:
Subtraction(-)
Addition(+)
Plus (+)
e. What is the value of e.
3
f. Be prepared to show your reasoning for your answers.
Compare your answer:
Although 3 and 9 are not zeros of , when is 3 and when is 9, is 21, or . Because the graphs of quadratic functions are symmetric, the -coordinate of the vertex is still 6. The -coordinate of the vertex is 3 because . The value of is 2 for the same reason as in function .
Video: Rewriting Expressions in Vertex Form
Watch the following video to learn more about rewriting expressions in vertex form.
Self Check
Additional Resources
Vertex Form with a Leading Coefficient Besides 1
Write in vertex form (of a quadratic expression).
Original expression
Step 1 - Rewrite the expression with 2 as a factor.
Step 2 - Add 4 to complete the square. Subtract 4 to keep the value of the expression the same.
Step 3 - Combine like terms so we can see the perfect square, .
Step 4 - Rewrite as a squared expression.
Step 5 - Apply the distributive property to put the expression in vertex form.
Try it
Try It: Vertex Form with a Leading Coefficient Besides 1
Write in vertex form.
Here is how to write the quadratic expression in vertex form:
Original expression
Step 1 - Rewrite the expression with 4 as a factor.
Step 2 - Add 9 to complete the square. Subtract 9 to keep the value of the expression the same.
Step 3 - Combine like terms so we can see the perfect square, .
Step 4 - Rewrite as a squared expression.
Step 5 - Apply the distributive property to put the expression in vertex form.