Find the Vertex -16t^2+20t+6

$- 16 t^{2} + 20 t + 6$

Step 1

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

$y = - 16 x^{2} + 20 x + 6$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- 16 x^{2} + 20 x + 6$ .

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

$a = - 16$

$b = 20$

$c = 6$

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

$d = \frac{20}{2 \cdot - 16}$

Simplify the right side.

Cancel the common factor of $20$ and $2$ .

Factor $2$ out of $20$ .

$d = \frac{2 \cdot 10}{2 \cdot - 16}$

Cancel the common factors.

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

$d = \frac{2 \cdot 10}{2 (- 16)}$

Cancel the common factor.

$d = \frac{2 \cdot 10}{2 \cdot - 16}$

Rewrite the expression.

$d = \frac{10}{- 16}$

Cancel the common factor of $10$ and $- 16$ .

Factor $2$ out of $10$ .

$d = \frac{2 (5)}{- 16}$

Cancel the common factors.

Factor $2$ out of $- 16$ .

$d = \frac{2 \cdot 5}{2 \cdot - 8}$

Cancel the common factor.

$d = \frac{2 \cdot 5}{2 \cdot - 8}$

Rewrite the expression.

$d = \frac{5}{- 8}$

Move the negative in front of the fraction.

$d = - \frac{5}{8}$

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

$e = 6 - \frac{20^{2}}{4 \cdot - 16}$

Simplify the right side.

Simplify each term.

Raise $20$ to the power of $2$ .

$e = 6 - \frac{400}{4 \cdot - 16}$

Multiply $4$ by $- 16$ .

$e = 6 - \frac{400}{- 64}$

Cancel the common factor of $400$ and $- 64$ .

Factor $16$ out of $400$ .

$e = 6 - \frac{16 (25)}{- 64}$

Cancel the common factors.

Factor $16$ out of $- 64$ .

$e = 6 - \frac{16 \cdot 25}{16 \cdot - 4}$

Cancel the common factor.

$e = 6 - \frac{16 \cdot 25}{16 \cdot - 4}$

Rewrite the expression.

$e = 6 - \frac{25}{- 4}$

Move the negative in front of the fraction.

$e = 6 - - \frac{25}{4}$

Multiply $- - \frac{25}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = 6 + 1 (\frac{25}{4})$

Multiply $\frac{25}{4}$ by $1$ .

$e = 6 + \frac{25}{4}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 6 \cdot \frac{4}{4} + \frac{25}{4}$

Combine $6$ and $\frac{4}{4}$ .

$e = \frac{6 \cdot 4}{4} + \frac{25}{4}$

Combine the numerators over the common denominator.

$e = \frac{6 \cdot 4 + 25}{4}$

Simplify the numerator.

Multiply $6$ by $4$ .

$e = \frac{24 + 25}{4}$

Add $24$ and $25$ .

$e = \frac{49}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 16 {(x - \frac{5}{8})}^{2} + \frac{49}{4}$ .

$- 16 {(x - \frac{5}{8})}^{2} + \frac{49}{4}$

Set $y$ equal to the new right side.

$y = - 16 {(x - \frac{5}{8})}^{2} + \frac{49}{4}$

Step 3

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

$a = - 16$

$h = \frac{5}{8}$

$k = \frac{49}{4}$

Step 4

Find the vertex $(h, k)$ .

$(\frac{5}{8}, \frac{49}{4})$

Step 5

Find the Vertex -16t^2+20t+6

Related Questions

Support LCMGCF.com by disabling your adblocker.