Find the Vertex h(x)=0.5(x+4)^2+1

$h (x) = 0.5 {(x + 4)}^{2} + 1$

Step 1

Rewrite the equation in vertex form.

Complete the square for $0.5 {(x + 4)}^{2} + 1$ .

Simplify the expression.

Simplify each term.

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$0.5 ((x + 4) (x + 4)) + 1$

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$0.5 (x (x + 4) + 4 (x + 4)) + 1$

Apply the distributive property.

$0.5 (x \cdot x + x \cdot 4 + 4 (x + 4)) + 1$

Apply the distributive property.

$0.5 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) + 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$0.5 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) + 1$

Move $4$ to the left of $x$ .

$0.5 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) + 1$

Multiply $4$ by $4$ .

$0.5 (x^{2} + 4 x + 4 x + 16) + 1$

Add $4 x$ and $4 x$ .

$0.5 (x^{2} + 8 x + 16) + 1$

Apply the distributive property.

$0.5 x^{2} + 0.5 (8 x) + 0.5 \cdot 16 + 1$

Simplify.

Multiply $8$ by $0.5$ .

$0.5 x^{2} + 4 x + 0.5 \cdot 16 + 1$

Multiply $0.5$ by $16$ .

$0.5 x^{2} + 4 x + 8 + 1$

Add $8$ and $1$ .

$0.5 x^{2} + 4 x + 9$

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

$a = 0.5$

$b = 4$

$c = 9$

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

$d = \frac{4}{2 \cdot 0.5}$

Simplify the right side.

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 2}{2 \cdot 0.5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 0.5$ .

$d = \frac{2 \cdot 2}{2 (0.5)}$

Cancel the common factor.

$d = \frac{2 \cdot 2}{2 \cdot 0.5}$

Rewrite the expression.

$d = \frac{2}{0.5}$

Divide $2$ by $0.5$ .

$d = 4$

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

$e = 9 - \frac{4^{2}}{4 \cdot 0.5}$

Simplify the right side.

Simplify each term.

Raise $4$ to the power of $2$ .

$e = 9 - \frac{16}{4 \cdot 0.5}$

Multiply $4$ by $0.5$ .

$e = 9 - \frac{16}{2}$

Divide $16$ by $2$ .

$e = 9 - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$e = 9 - 8$

Subtract $8$ from $9$ .

$e = 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $0.5 {(x + 4)}^{2} + 1$ .

$0.5 {(x + 4)}^{2} + 1$

Set $y$ equal to the new right side.

$y = 0.5 {(x + 4)}^{2} + 1$

Step 2

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

$a = 0.5$

$h = - 4$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(- 4, 1)$

Step 4

Find the Vertex h(x)=0.5(x+4)^2+1

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