Activity
For questions 1 – 5, write each function in vertex form.
Compare your answer:
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For questions 6 and 7, write each function in standard form.
Compare your answer:
Compare your answer:
Are you ready for more?
Extending Your Thinking
A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation . What is the maximum height?
Compare your answer:
To find the maximum height, find the vertex. Write the function in vertex form.
Step 1 - Factor -16 out of the first two terms.
Step 2 - Complete the square.
Step 3 - Add the constant to complete the square and subtract from the 40 to balance the equation. Note that because there is a outside of the parentheses. This makes the constant equation to be . Because a was really added in, then a positive 100 will balance the equation.
Step 4 - Factor and simplify.
Step 5 - Find the vertex.
Therefore, the maximum height is 140 feet. (It reaches this high point after 2.5 seconds.)
Self Check
Additional Resources
Writing Quadratics in Different Forms
Three Common Quadratic Forms:
The standard form of a quadratic function is where , , , are real numbers and .
The vertex form of a quadratic function is where is the vertex.
The factored form of the quadratic function is where and are roots of .
Example 1
Write the function in standard form.
Step 1 - multiply the binomials using distribution (FOIL).
Step 2 - Simplify.
Example 2
Write the function in vertex form.
Hint: Start by using the answer from example 1.
Step 1 - Complete the square.
Step 2 - Balance the equation. Add the constant to the quadratic to complete the square and subtract it from the original constant.
Finally, factor the trinomial and simplify.
Example 3
Rewrite the function in standard form:
Step 1 - Expand the square.
Step 2 - Distribute the coefficient.
Step 3 - Simplify like terms
Example 4
Rewrite the function in standard form:
Step 1 - Expand the square.
Step 2 - Distribute the coefficient.
Step 3 - Simplify like terms
Try it
Try It: Writing Quadratic Functions in Equivalent Forms
Write the function in vertex form.
Compare your answer:
First, multiply the binomials using distribution (FOIL).
Then, simplify.
Now, write this function in vertex form. Begin by completing the square.
Balance the equation. Add the constant to the quadratic to complete the square and subtract it from the original constant.
Finally, factor the trinomial and simplify.