Find the Vertex -16t^2+24t+7

$- 16 t^{2} + 24 t + 7$

Step 1

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

$y = - 16 x^{2} + 24 x + 7$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- 16 x^{2} + 24 x + 7$ .

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

$a = - 16$

$b = 24$

$c = 7$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{24}{2 \cdot - 16}$

Simplify the right side.

Cancel the common factor of $24$ and $2$ .

Factor $2$ out of $24$ .

$d = \frac{2 \cdot 12}{2 \cdot - 16}$

Cancel the common factors.

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

$d = \frac{2 \cdot 12}{2 (- 16)}$

Cancel the common factor.

$d = \frac{2 \cdot 12}{2 \cdot - 16}$

Rewrite the expression.

$d = \frac{12}{- 16}$

Cancel the common factor of $12$ and $- 16$ .

Factor $4$ out of $12$ .

$d = \frac{4 (3)}{- 16}$

Cancel the common factors.

Factor $4$ out of $- 16$ .

$d = \frac{4 \cdot 3}{4 \cdot - 4}$

Cancel the common factor.

$d = \frac{4 \cdot 3}{4 \cdot - 4}$

Rewrite the expression.

$d = \frac{3}{- 4}$

Move the negative in front of the fraction.

$d = - \frac{3}{4}$

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

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

$e = 7 - \frac{24^{2}}{4 \cdot - 16}$

Simplify the right side.

Simplify each term.

Raise $24$ to the power of $2$ .

$e = 7 - \frac{576}{4 \cdot - 16}$

Multiply $4$ by $- 16$ .

$e = 7 - \frac{576}{- 64}$

Divide $576$ by $- 64$ .

$e = 7 - - 9$

Multiply $- 1$ by $- 9$ .

$e = 7 + 9$

Add $7$ and $9$ .

$e = 16$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 16 {(x - \frac{3}{4})}^{2} + 16$ .

$- 16 {(x - \frac{3}{4})}^{2} + 16$

Set $y$ equal to the new right side.

$y = - 16 {(x - \frac{3}{4})}^{2} + 16$

Step 3

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

$a = - 16$

$h = \frac{3}{4}$

$k = 16$

Step 4

Find the vertex $(h, k)$ .

$(\frac{3}{4}, 16)$

Step 5

Find the Vertex -16t^2+24t+7

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