Find the Vertex 56t-16t^2

$56 t - 16 t^{2}$

Step 1

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

$y = 56 x - 16 x^{2}$

Step 2

Rewrite the equation in vertex form.

Reorder $56 x$ and $- 16 x^{2}$ .

$y = - 16 x^{2} + 56 x$

Complete the square for $- 16 x^{2} + 56 x$ .

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

$a = - 16$

$b = 56$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{56}{2 \cdot - 16}$

Simplify the right side.

Cancel the common factor of $56$ and $2$ .

Factor $2$ out of $56$ .

$d = \frac{2 \cdot 28}{2 \cdot - 16}$

Cancel the common factors.

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

$d = \frac{2 \cdot 28}{2 (- 16)}$

Cancel the common factor.

$d = \frac{2 \cdot 28}{2 \cdot - 16}$

Rewrite the expression.

$d = \frac{28}{- 16}$

Cancel the common factor of $28$ and $- 16$ .

Factor $4$ out of $28$ .

$d = \frac{4 (7)}{- 16}$

Cancel the common factors.

Factor $4$ out of $- 16$ .

$d = \frac{4 \cdot 7}{4 \cdot - 4}$

Cancel the common factor.

$d = \frac{4 \cdot 7}{4 \cdot - 4}$

Rewrite the expression.

$d = \frac{7}{- 4}$

Move the negative in front of the fraction.

$d = - \frac{7}{4}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{56^{2}}{4 \cdot - 16}$

Simplify the right side.

Simplify each term.

Raise $56$ to the power of $2$ .

$e = 0 - \frac{3136}{4 \cdot - 16}$

Multiply $4$ by $- 16$ .

$e = 0 - \frac{3136}{- 64}$

Divide $3136$ by $- 64$ .

$e = 0 - - 49$

Multiply $- 1$ by $- 49$ .

$e = 0 + 49$

Add $0$ and $49$ .

$e = 49$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 16 {(x - \frac{7}{4})}^{2} + 49$ .

$- 16 {(x - \frac{7}{4})}^{2} + 49$

Set $y$ equal to the new right side.

$y = - 16 {(x - \frac{7}{4})}^{2} + 49$

Step 3

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

$a = - 16$

$h = \frac{7}{4}$

$k = 49$

Step 4

Find the vertex $(h, k)$ .

$(\frac{7}{4}, 49)$

Step 5

Find the Vertex 56t-16t^2

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