Algebra Examples

$y = - \frac{1}{5} {(x + 5)}^{2} + 2$

Step 1

The parent function is the simplest form of the type of function given.

$y = x^{2}$

Step 2

Simplify $- \frac{1}{5} \cdot {(x + 5)}^{2} + 2$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$y = - \frac{1}{5} \cdot ((x + 5) (x + 5)) + 2$

Step 2.1.2

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

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Step 2.1.2.1

Apply the distributive property.

$y = - \frac{1}{5} \cdot (x (x + 5) + 5 (x + 5)) + 2$

Step 2.1.2.2

Apply the distributive property.

$y = - \frac{1}{5} \cdot (x \cdot x + x \cdot 5 + 5 (x + 5)) + 2$

Step 2.1.2.3

Apply the distributive property.

$y = - \frac{1}{5} \cdot (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) + 2$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

$y = - \frac{1}{5} \cdot (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) + 2$

Step 2.1.3.1.2

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

$y = - \frac{1}{5} \cdot (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) + 2$

Step 2.1.3.1.3

Multiply $5$ by $5$ .

$y = - \frac{1}{5} \cdot (x^{2} + 5 x + 5 x + 25) + 2$

Step 2.1.3.2

Add $5 x$ and $5 x$ .

$y = - \frac{1}{5} \cdot (x^{2} + 10 x + 25) + 2$

Step 2.1.4

Apply the distributive property.

$y = - \frac{1}{5} x^{2} - \frac{1}{5} (10 x) - \frac{1}{5} \cdot 25 + 2$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Combine $x^{2}$ and $\frac{1}{5}$ .

$y = - \frac{x^{2}}{5} - \frac{1}{5} (10 x) - \frac{1}{5} \cdot 25 + 2$

Step 2.1.5.2

Cancel the common factor of $5$ .

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Step 2.1.5.2.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$y = - \frac{x^{2}}{5} + \frac{- 1}{5} (10 x) - \frac{1}{5} \cdot 25 + 2$

Step 2.1.5.2.2

Factor $5$ out of $10 x$ .

$y = - \frac{x^{2}}{5} + \frac{- 1}{5} (5 (2 x)) - \frac{1}{5} \cdot 25 + 2$

Step 2.1.5.2.3

Cancel the common factor.

$y = - \frac{x^{2}}{5} + \frac{- 1}{5} (5 (2 x)) - \frac{1}{5} \cdot 25 + 2$

Step 2.1.5.2.4

Rewrite the expression.

$y = - \frac{x^{2}}{5} - 1 (2 x) - \frac{1}{5} \cdot 25 + 2$

Step 2.1.5.3

Multiply $2$ by $- 1$ .

$y = - \frac{x^{2}}{5} - 2 x - \frac{1}{5} \cdot 25 + 2$

Step 2.1.5.4

Cancel the common factor of $5$ .

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Step 2.1.5.4.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$y = - \frac{x^{2}}{5} - 2 x + \frac{- 1}{5} \cdot 25 + 2$

Step 2.1.5.4.2

Factor $5$ out of $25$ .

$y = - \frac{x^{2}}{5} - 2 x + \frac{- 1}{5} \cdot (5 (5)) + 2$

Step 2.1.5.4.3

Cancel the common factor.

$y = - \frac{x^{2}}{5} - 2 x + \frac{- 1}{5} \cdot (5 \cdot 5) + 2$

Step 2.1.5.4.4

Rewrite the expression.

$y = - \frac{x^{2}}{5} - 2 x - 1 \cdot 5 + 2$

Step 2.1.5.5

Multiply $- 1$ by $5$ .

$y = - \frac{x^{2}}{5} - 2 x - 5 + 2$

Step 2.2

Add $- 5$ and $2$ .

$y = - \frac{x^{2}}{5} - 2 x - 3$

Step 3

Assume that $y = x^{2}$ is $f (x) = x^{2}$ and $y = - \frac{1}{5} \cdot {(x + 5)}^{2} + 2$ is $g (x) = - \frac{x^{2}}{5} - 2 x - 3$ .

$f (x) = x^{2}$

$g (x) = - \frac{x^{2}}{5} - 2 x - 3$

Step 4

The transformation being described is from $f (x) = x^{2}$ to $g (x) = - \frac{x^{2}}{5} - 2 x - 3$ .

$f (x) = x^{2} \to g (x) = - \frac{x^{2}}{5} - 2 x - 3$

Step 5

Find the vertex form of $g (x) = - \frac{x^{2}}{5} - 2 x - 3$ .

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Step 5.1

Complete the square for $- \frac{x^{2}}{5} - 2 x - 3$ .

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Step 5.1.1

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

$a = - \frac{1}{5}$

$b = - 2$

$c = - 3$

Step 5.1.2

Consider the vertex form of a parabola.

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

Step 5.1.3

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

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Step 5.1.3.1

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

$d = \frac{- 2}{2 (- \frac{1}{5})}$

Step 5.1.3.2

Simplify the right side.

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Step 5.1.3.2.1

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

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Step 5.1.3.2.1.1

Factor $2$ out of $- 2$ .

$d = \frac{2 \cdot - 1}{2 (- \frac{1}{5})}$

Step 5.1.3.2.1.2

Cancel the common factors.

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Step 5.1.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot - 1}{2 (- \frac{1}{5})}$

Step 5.1.3.2.1.2.2

Rewrite the expression.

$d = \frac{- 1}{- \frac{1}{5}}$

Step 5.1.3.2.2

Dividing two negative values results in a positive value.

$d = \frac{1}{\frac{1}{5}}$

Step 5.1.3.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 5$

Step 5.1.3.2.4

Multiply $5$ by $1$ .

$d = 5$

Step 5.1.4

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

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Step 5.1.4.1

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

$e = - 3 - \frac{{(- 2)}^{2}}{4 (- \frac{1}{5})}$

Step 5.1.4.2

Simplify the right side.

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Step 5.1.4.2.1

Simplify each term.

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Step 5.1.4.2.1.1

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

$e = - 3 - \frac{4}{4 (- \frac{1}{5})}$

Step 5.1.4.2.1.2

Simplify the denominator.

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Step 5.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - 3 - \frac{4}{- 4 (\frac{1}{5})}$

Step 5.1.4.2.1.2.2

Combine $- 4$ and $\frac{1}{5}$ .

$e = - 3 - \frac{4}{\frac{- 4}{5}}$

Step 5.1.4.2.1.3

Move the negative in front of the fraction.

$e = - 3 - \frac{4}{- \frac{4}{5}}$

Step 5.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - 3 - (4 (- \frac{5}{4}))$

Step 5.1.4.2.1.5

Cancel the common factor of $4$ .

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Step 5.1.4.2.1.5.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$e = - 3 - (4 (\frac{- 5}{4}))$

Step 5.1.4.2.1.5.2

Cancel the common factor.

$e = - 3 - (4 (\frac{- 5}{4}))$

Step 5.1.4.2.1.5.3

Rewrite the expression.

$e = - 3 - - 5$

Step 5.1.4.2.1.6

Multiply $- 1$ by $- 5$ .

$e = - 3 + 5$

Step 5.1.4.2.2

Add $- 3$ and $5$ .

$e = 2$

Step 5.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{5} {(x + 5)}^{2} + 2$ .

$- \frac{1}{5} {(x + 5)}^{2} + 2$

Step 5.2

Set $y$ equal to the new right side.

$y = - \frac{1}{5} \cdot {(x + 5)}^{2} + 2$

Step 6

The horizontal shift depends on the value of $h$ . The horizontal shift is described as:

$g (x) = f (x + h)$ - The graph is shifted to the left $h$ units.

$g (x) = f (x - h)$ - The graph is shifted to the right $h$ units.

Horizontal Shift: Left $5$ Units

Step 7

The vertical shift depends on the value of $k$ . The vertical shift is described as:

$g (x) = f (x) + k$ - The graph is shifted up $k$ units.

$g (x) = f (x) - k$ - The graph is shifted down $k$ units.

Vertical Shift: Up $2$ Units

Step 8

The graph is reflected about the x-axis when $g (x) = - f (x)$ .

Reflection about the x-axis: Reflected

Step 9

The graph is reflected about the y-axis when $g (x) = f (- x)$ .

Reflection about the y-axis: None

Step 10

Compressing and stretching depends on the value of $a$ .

When $a$ is greater than $1$ : Vertically stretched

When $a$ is between $0$ and $1$ : Vertically compressed

Vertical Compression or Stretch: Compressed

Step 11

Compare and list the transformations.

Parent Function: $y = x^{2}$

Horizontal Shift: Left $5$ Units

Vertical Shift: Up $2$ Units

Reflection about the x-axis: Reflected

Reflection about the y-axis: None

Vertical Compression or Stretch: Compressed

Step 12