Question

question 2
question
this equation models the height over time of a deep-
h(t) = -4.9t² + 29.4t + 5.9
rewrite the function in vertex form.
enter the correct answer in the box.

h(t) =

Explanation:

Step1: Factor out the coefficient of \( t^2 \)

We have the quadratic function \( h(t) = -4.9t^2 + 29.4t + 5.9 \). First, factor out \(-4.9\) from the first two terms:
\( h(t) = -4.9(t^2 - 6t) + 5.9 \) (since \( \frac{29.4}{-4.9} = -6 \))

Step2: Complete the square inside the parentheses

To complete the square for \( t^2 - 6t \), we take half of the coefficient of \( t \), which is \(\frac{-6}{2} = -3\), square it: \( (-3)^2 = 9 \). So we add and subtract 9 inside the parentheses (but we have to be careful with the factored out coefficient):
\( h(t) = -4.9(t^2 - 6t + 9 - 9) + 5.9 \)
Rewrite the expression inside the parentheses as a perfect square and a constant:
\( h(t) = -4.9((t - 3)^2 - 9) + 5.9 \)

Step3: Distribute the -4.9

Distribute \(-4.9\) to both terms inside the parentheses:
\( h(t) = -4.9(t - 3)^2 + (-4.9)(-9) + 5.9 \)
Calculate \( (-4.9)(-9) = 44.1 \)

Step4: Combine the constant terms

Now combine \( 44.1 \) and \( 5.9 \):
\( 44.1 + 5.9 = 50 \)
So the function in vertex form is:
\( h(t) = -4.9(t - 3)^2 + 50 \)

Answer:

\( h(t) = -4.9(t - 3)^2 + 50 \)