Precalculus Examples

$f (x) = 0.1 {(x - 3.2)}^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $0.1 {(x - 3.2)}^{2}$ .

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Step 1.1.1

Simplify the expression.

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Step 1.1.1.1

Rewrite ${(x - 3.2)}^{2}$ as $(x - 3.2) (x - 3.2)$ .

$0.1 ((x - 3.2) (x - 3.2))$

Step 1.1.1.2

Expand $(x - 3.2) (x - 3.2)$ using the FOIL Method.

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Step 1.1.1.2.1

Apply the distributive property.

$0.1 (x (x - 3.2) - 3.2 (x - 3.2))$

Step 1.1.1.2.2

Apply the distributive property.

$0.1 (x \cdot x + x \cdot - 3.2 - 3.2 (x - 3.2))$

Step 1.1.1.2.3

Apply the distributive property.

$0.1 (x \cdot x + x \cdot - 3.2 - 3.2 x - 3.2 \cdot - 3.2)$

Step 1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.3.1

Simplify each term.

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Step 1.1.1.3.1.1

Multiply $x$ by $x$ .

$0.1 (x^{2} + x \cdot - 3.2 - 3.2 x - 3.2 \cdot - 3.2)$

Step 1.1.1.3.1.2

Move $- 3.2$ to the left of $x$ .

$0.1 (x^{2} - 3.2 \cdot x - 3.2 x - 3.2 \cdot - 3.2)$

Step 1.1.1.3.1.3

Multiply $- 3.2$ by $- 3.2$ .

$0.1 (x^{2} - 3.2 x - 3.2 x + 10.24)$

Step 1.1.1.3.2

Subtract $3.2 x$ from $- 3.2 x$ .

$0.1 (x^{2} - 6.4 x + 10.24)$

Step 1.1.1.4

Apply the distributive property.

$0.1 x^{2} + 0.1 (- 6.4 x) + 0.1 \cdot 10.24$

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

Multiply $- 6.4$ by $0.1$ .

$0.1 x^{2} - 0.64 x + 0.1 \cdot 10.24$

Step 1.1.1.5.2

Multiply $0.1$ by $10.24$ .

$0.1 x^{2} - 0.64 x + 1.024$

Step 1.1.2

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

$a = 0.1$

$b = - 0.64$

$c = 1.024$

Step 1.1.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 0.64}{2 \cdot 0.1}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Multiply $2$ by $0.1$ .

$d = \frac{- 0.64}{0.2}$

Step 1.1.4.2.2

Divide $- 0.64$ by $0.2$ .

$d = - 3.2$

Step 1.1.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 1.024 - \frac{{(- 0.64)}^{2}}{4 \cdot 0.1}$

Step 1.1.5.2

Simplify the right side.

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Step 1.1.5.2.1

Simplify each term.

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Step 1.1.5.2.1.1

Raise $- 0.64$ to the power of $2$ .

$e = 1.024 - \frac{0.4096}{4 \cdot 0.1}$

Step 1.1.5.2.1.2

Multiply $4$ by $0.1$ .

$e = 1.024 - \frac{0.4096}{0.4}$

Step 1.1.5.2.1.3

Divide $0.4096$ by $0.4$ .

$e = 1.024 - 1 \cdot 1.024$

Step 1.1.5.2.1.4

Multiply $- 1$ by $1.024$ .

$e = 1.024 - 1.024$

Step 1.1.5.2.2

Subtract $1.024$ from $1.024$ .

$e = 0$

Step 1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $0.1 {(x - 3.2)}^{2} + 0$ .

$0.1 {(x - 3.2)}^{2} + 0$

Step 1.2

Set $y$ equal to the new right side.

$y = 0.1 {(x - 3.2)}^{2} + 0$

Step 2

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

$a = 0.1$

$h = 3.2$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(3.2, 0)$

Step 4