Algebra Examples

$f (x) = - 0.2 {(x + 3)}^{2} + 2$

Step 1

Write $f (x) = - 0.2 {(x + 3)}^{2} + 2$ as an equation.

$y = - 0.2 {(x + 3)}^{2} + 2$

Step 2

Rewrite the equation in vertex form.

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Step 2.1

Complete the square for $- 0.2 {(x + 3)}^{2} + 2$ .

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Step 2.1.1

Simplify the expression.

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Step 2.1.1.1

Simplify each term.

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Step 2.1.1.1.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$- 0.2 ((x + 3) (x + 3)) + 2$

Step 2.1.1.1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 2.1.1.1.2.1

Apply the distributive property.

$- 0.2 (x (x + 3) + 3 (x + 3)) + 2$

Step 2.1.1.1.2.2

Apply the distributive property.

$- 0.2 (x \cdot x + x \cdot 3 + 3 (x + 3)) + 2$

Step 2.1.1.1.2.3

Apply the distributive property.

$- 0.2 (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + 2$

Step 2.1.1.1.3

Simplify and combine like terms.

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Step 2.1.1.1.3.1

Simplify each term.

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Step 2.1.1.1.3.1.1

Multiply $x$ by $x$ .

$- 0.2 (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) + 2$

Step 2.1.1.1.3.1.2

Move $3$ to the left of $x$ .

$- 0.2 (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) + 2$

Step 2.1.1.1.3.1.3

Multiply $3$ by $3$ .

$- 0.2 (x^{2} + 3 x + 3 x + 9) + 2$

Step 2.1.1.1.3.2

Add $3 x$ and $3 x$ .

$- 0.2 (x^{2} + 6 x + 9) + 2$

Step 2.1.1.1.4

Apply the distributive property.

$- 0.2 x^{2} - 0.2 (6 x) - 0.2 \cdot 9 + 2$

Step 2.1.1.1.5

Simplify.

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Step 2.1.1.1.5.1

Multiply $6$ by $- 0.2$ .

$- 0.2 x^{2} - 1.2 x - 0.2 \cdot 9 + 2$

Step 2.1.1.1.5.2

Multiply $- 0.2$ by $9$ .

$- 0.2 x^{2} - 1.2 x - 1.8 + 2$

Step 2.1.1.2

Add $- 1.8$ and $2$ .

$- 0.2 x^{2} - 1.2 x + 0.2$

Step 2.1.2

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

$a = - 0.2$

$b = - 1.2$

$c = 0.2$

Step 2.1.3

Consider the vertex form of a parabola.

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

Step 2.1.4

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

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Step 2.1.4.1

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

$d = \frac{- 1.2}{2 \cdot - 0.2}$

Step 2.1.4.2

Simplify the right side.

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Step 2.1.4.2.1

Multiply $2$ by $- 0.2$ .

$d = \frac{- 1.2}{- 0.4}$

Step 2.1.4.2.2

Divide $- 1.2$ by $- 0.4$ .

$d = 3$

Step 2.1.5

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

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Step 2.1.5.1

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

$e = 0.2 - \frac{{(- 1.2)}^{2}}{4 \cdot - 0.2}$

Step 2.1.5.2

Simplify the right side.

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Step 2.1.5.2.1

Simplify each term.

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Step 2.1.5.2.1.1

Raise $- 1.2$ to the power of $2$ .

$e = 0.2 - \frac{1.44}{4 \cdot - 0.2}$

Step 2.1.5.2.1.2

Multiply $4$ by $- 0.2$ .

$e = 0.2 - \frac{1.44}{- 0.8}$

Step 2.1.5.2.1.3

Divide $1.44$ by $- 0.8$ .

$e = 0.2 - - 1.8$

Step 2.1.5.2.1.4

Multiply $- 1$ by $- 1.8$ .

$e = 0.2 + 1.8$

Step 2.1.5.2.2

Add $0.2$ and $1.8$ .

$e = 2$

Step 2.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 0.2 {(x + 3)}^{2} + 2$ .

$- 0.2 {(x + 3)}^{2} + 2$

Step 2.2

Set $y$ equal to the new right side.

$y = - 0.2 {(x + 3)}^{2} + 2$

Step 3

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

$a = - 0.2$

$h = - 3$

$k = 2$

Step 4

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- 3, 2)$

Step 6

Find $p$ , the distance from the vertex to the focus.

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Step 6.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 6.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 0.2}$

Step 6.3

Simplify.

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Step 6.3.1

Multiply $4$ by $- 0.2$ .

$\frac{1}{- 0.8}$

Step 6.3.2

Divide $1$ by $- 0.8$ .

$- 1.25$

Step 7

Find the focus.

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Step 7.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 7.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- 3, 0.75)$

Step 8

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = - 3$

Step 9