Find the Vertex y=0.25(x+5)^2

$y = 0.25 {(x + 5)}^{2}$

Step 1

Rewrite the equation in vertex form.

Complete the square for $0.25 {(x + 5)}^{2}$ .

Simplify the expression.

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$0.25 ((x + 5) (x + 5))$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$0.25 (x (x + 5) + 5 (x + 5))$

Apply the distributive property.

$0.25 (x \cdot x + x \cdot 5 + 5 (x + 5))$

Apply the distributive property.

$0.25 (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$0.25 (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5)$

Move $5$ to the left of $x$ .

$0.25 (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5)$

Multiply $5$ by $5$ .

$0.25 (x^{2} + 5 x + 5 x + 25)$

Add $5 x$ and $5 x$ .

$0.25 (x^{2} + 10 x + 25)$

Apply the distributive property.

$0.25 x^{2} + 0.25 (10 x) + 0.25 \cdot 25$

Simplify.

Multiply $10$ by $0.25$ .

$0.25 x^{2} + 2.5 x + 0.25 \cdot 25$

Multiply $0.25$ by $25$ .

$0.25 x^{2} + 2.5 x + 6.25$

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

$a = 0.25$

$b = 2.5$

$c = 6.25$

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

$d = \frac{2.5}{2 \cdot 0.25}$

Simplify the right side.

Multiply $2$ by $0.25$ .

$d = \frac{2.5}{0.5}$

Divide $2.5$ by $0.5$ .

$d = 5$

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

$e = 6.25 - \frac{{2.5}^{2}}{4 \cdot 0.25}$

Simplify the right side.

Simplify each term.

Raise $2.5$ to the power of $2$ .

$e = 6.25 - \frac{6.25}{4 \cdot 0.25}$

Multiply $4$ by $0.25$ .

$e = 6.25 - \frac{6.25}{1}$

Divide $6.25$ by $1$ .

$e = 6.25 - 1 \cdot 6.25$

Multiply $- 1$ by $6.25$ .

$e = 6.25 - 6.25$

Subtract $6.25$ from $6.25$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $0.25 {(x + 5)}^{2} + 0$ .

$0.25 {(x + 5)}^{2} + 0$

Set $y$ equal to the new right side.

$y = 0.25 {(x + 5)}^{2} + 0$

Step 2

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

$a = 0.25$

$h = - 5$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(- 5, 0)$

Step 4

Find the Vertex y=0.25(x+5)^2

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