Precalculus Examples

$h (t) = - 16 t^{2} + 128 t$

Step 1

Rewrite the equation in terms of $x$ and $y$ .

$y = - 16 x^{2} + 128 x$

Step 2

Rewrite the equation in vertex form.

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Step 2.1

Complete the square for $- 16 x^{2} + 128 x$ .

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Step 2.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 16$

$b = 128$

$c = 0$

Step 2.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 2.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{128}{2 \cdot - 16}$

Step 2.1.3.2

Simplify the right side.

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Step 2.1.3.2.1

Cancel the common factor of $128$ and $2$ .

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Step 2.1.3.2.1.1

Factor $2$ out of $128$ .

$d = \frac{2 \cdot 64}{2 \cdot - 16}$

Step 2.1.3.2.1.2

Cancel the common factors.

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Step 2.1.3.2.1.2.1

Factor $2$ out of $2 \cdot - 16$ .

$d = \frac{2 \cdot 64}{2 (- 16)}$

Step 2.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 64}{2 \cdot - 16}$

Step 2.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{64}{- 16}$

Step 2.1.3.2.2

Cancel the common factor of $64$ and $- 16$ .

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Step 2.1.3.2.2.1

Factor $16$ out of $64$ .

$d = \frac{16 (4)}{- 16}$

Step 2.1.3.2.2.2

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$d = - 1 \cdot 4$

Step 2.1.3.2.3

Multiply $- 1$ by $4$ .

$d = - 4$

Step 2.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{128^{2}}{4 \cdot - 16}$

Step 2.1.4.2

Simplify the right side.

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Step 2.1.4.2.1

Simplify each term.

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Step 2.1.4.2.1.1

Raise $128$ to the power of $2$ .

$e = 0 - \frac{16384}{4 \cdot - 16}$

Step 2.1.4.2.1.2

Multiply $4$ by $- 16$ .

$e = 0 - \frac{16384}{- 64}$

Step 2.1.4.2.1.3

Divide $16384$ by $- 64$ .

$e = 0 - - 256$

Step 2.1.4.2.1.4

Multiply $- 1$ by $- 256$ .

$e = 0 + 256$

Step 2.1.4.2.2

Add $0$ and $256$ .

$e = 256$

Step 2.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 16 {(x - 4)}^{2} + 256$ .

$- 16 {(x - 4)}^{2} + 256$

Step 2.2

Set $y$ equal to the new right side.

$y = - 16 {(x - 4)}^{2} + 256$

Step 3

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 16$

$h = 4$

$k = 256$

Step 4

Find the vertex $(h, k)$ .

$(4, 256)$

Step 5