Find the Vertex -4.9t^2+14.7t+1

$- 4.9 t^{2} + 14.7 t + 1$

Step 1

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

$y = - 4.9 x^{2} + 14.7 x + 1$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- 4.9 x^{2} + 14.7 x + 1$ .

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

$a = - 4.9$

$b = 14.7$

$c = 1$

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

$d = \frac{14.7}{2 \cdot - 4.9}$

Simplify the right side.

Multiply $2$ by $- 4.9$ .

$d = \frac{14.7}{- 9.8}$

Divide $14.7$ by $- 9.8$ .

$d = - 1.5$

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

$e = 1 - \frac{{14.7}^{2}}{4 \cdot - 4.9}$

Simplify the right side.

Simplify each term.

Raise $14.7$ to the power of $2$ .

$e = 1 - \frac{216.09}{4 \cdot - 4.9}$

Multiply $4$ by $- 4.9$ .

$e = 1 - \frac{216.09}{- 19.6}$

Divide $216.09$ by $- 19.6$ .

$e = 1 - - 11.025$

Multiply $- 1$ by $- 11.025$ .

$e = 1 + 11.025$

Add $1$ and $11.025$ .

$e = 12.025$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4.9 {(x - 1.5)}^{2} + 12.025$ .

$- 4.9 {(x - 1.5)}^{2} + 12.025$

Set $y$ equal to the new right side.

$y = - 4.9 {(x - 1.5)}^{2} + 12.025$

Step 3

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

$a = - 4.9$

$h = 1.5$

$k = 12.025$

Step 4

Find the vertex $(h, k)$ .

$(1.5, 12.025)$

Step 5

Find the Vertex -4.9t^2+14.7t+1

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