Find the Vertex f(r)=(r-14)^2-56.25

$f (r) = {(r - 14)}^{2} - 56.25$

Step 1

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

$y = {(x - 14)}^{2} - 56.25$

Step 2

Rewrite the equation in vertex form.

Complete the square for ${(x - 14)}^{2} - 56.25$ .

Simplify the expression.

Simplify each term.

Rewrite ${(x - 14)}^{2}$ as $(x - 14) (x - 14)$ .

$(x - 14) (x - 14) - 56.25$

Expand $(x - 14) (x - 14)$ using the FOIL Method.

Apply the distributive property.

$x (x - 14) - 14 (x - 14) - 56.25$

Apply the distributive property.

$x \cdot x + x \cdot - 14 - 14 (x - 14) - 56.25$

Apply the distributive property.

$x \cdot x + x \cdot - 14 - 14 x - 14 \cdot - 14 - 56.25$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 14 - 14 x - 14 \cdot - 14 - 56.25$

Move $- 14$ to the left of $x$ .

$x^{2} - 14 \cdot x - 14 x - 14 \cdot - 14 - 56.25$

Multiply $- 14$ by $- 14$ .

$x^{2} - 14 x - 14 x + 196 - 56.25$

Subtract $14 x$ from $- 14 x$ .

$x^{2} - 28 x + 196 - 56.25$

Subtract $56.25$ from $196$ .

$x^{2} - 28 x + 139.75$

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

$a = 1$

$b = - 28$

$c = 139.75$

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

$d = \frac{- 28}{2 \cdot 1}$

Cancel the common factor of $- 28$ and $2$ .

Factor $2$ out of $- 28$ .

$d = \frac{2 \cdot - 14}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 14}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 14}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 14}{1}$

Divide $- 14$ by $1$ .

$d = - 14$

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

$e = 139.75 - \frac{{(- 28)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 28$ to the power of $2$ .

$e = 139.75 - \frac{784}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 139.75 - \frac{784}{4}$

Divide $784$ by $4$ .

$e = 139.75 - 1 \cdot 196$

Multiply $- 1$ by $196$ .

$e = 139.75 - 196$

Subtract $196$ from $139.75$ .

$e = - 56.25$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 14)}^{2} - 56.25$ .

${(x - 14)}^{2} - 56.25$

Set $y$ equal to the new right side.

$y = {(x - 14)}^{2} - 56.25$

Step 3

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

$a = 1$

$h = 14$

$k = - 56.25$

Step 4

Find the vertex $(h, k)$ .

$(14, - 56.25)$

Step 5

Find the Vertex f(r)=(r-14)^2-56.25

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