Algebra Examples

$- {(x - 5)}^{2} + 3$

Step 1

Write $- {(x - 5)}^{2} + 3$ as an equation.

$y = - {(x - 5)}^{2} + 3$

Step 2

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

$y = x^{2}$

Step 3

Simplify $- {(x - 5)}^{2} + 3$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$y = - ((x - 5) (x - 5)) + 3$

Step 3.1.2

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

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Step 3.1.2.1

Apply the distributive property.

$y = - (x (x - 5) - 5 (x - 5)) + 3$

Step 3.1.2.2

Apply the distributive property.

$y = - (x \cdot x + x \cdot - 5 - 5 (x - 5)) + 3$

Step 3.1.2.3

Apply the distributive property.

$y = - (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) + 3$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply $x$ by $x$ .

$y = - (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) + 3$

Step 3.1.3.1.2

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

$y = - (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) + 3$

Step 3.1.3.1.3

Multiply $- 5$ by $- 5$ .

$y = - (x^{2} - 5 x - 5 x + 25) + 3$

Step 3.1.3.2

Subtract $5 x$ from $- 5 x$ .

$y = - (x^{2} - 10 x + 25) + 3$

Step 3.1.4

Apply the distributive property.

$y = - x^{2} - (- 10 x) - 1 \cdot 25 + 3$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Multiply $- 10$ by $- 1$ .

$y = - x^{2} + 10 x - 1 \cdot 25 + 3$

Step 3.1.5.2

Multiply $- 1$ by $25$ .

$y = - x^{2} + 10 x - 25 + 3$

Step 3.2

Add $- 25$ and $3$ .

$y = - x^{2} + 10 x - 22$

Step 4

Assume that $y = x^{2}$ is $f (x) = x^{2}$ and $y = - {(x - 5)}^{2} + 3$ is $g (x) = - x^{2} + 10 x - 22$ .

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

$g (x) = - x^{2} + 10 x - 22$

Step 5

The transformation being described is from $f (x) = x^{2}$ to $g (x) = - x^{2} + 10 x - 22$ .

$f (x) = x^{2} \to g (x) = - x^{2} + 10 x - 22$

Step 6

Find the vertex form of $g (x) = - x^{2} + 10 x - 22$ .

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Step 6.1

Complete the square for $- x^{2} + 10 x - 22$ .

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Step 6.1.1

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

$a = - 1$

$b = 10$

$c = - 22$

Step 6.1.2

Consider the vertex form of a parabola.

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

Step 6.1.3

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

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Step 6.1.3.1

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

$d = \frac{10}{2 \cdot - 1}$

Step 6.1.3.2

Simplify the right side.

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Step 6.1.3.2.1

Cancel the common factor of $10$ and $2$ .

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Step 6.1.3.2.1.1

Factor $2$ out of $10$ .

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

Step 6.1.3.2.1.2

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Step 6.1.3.2.2

Multiply $- 1$ by $5$ .

$d = - 5$

Step 6.1.4

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

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Step 6.1.4.1

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

$e = - 22 - \frac{10^{2}}{4 \cdot - 1}$

Step 6.1.4.2

Simplify the right side.

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Step 6.1.4.2.1

Simplify each term.

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Step 6.1.4.2.1.1

Raise $10$ to the power of $2$ .

$e = - 22 - \frac{100}{4 \cdot - 1}$

Step 6.1.4.2.1.2

Multiply $4$ by $- 1$ .

$e = - 22 - \frac{100}{- 4}$

Step 6.1.4.2.1.3

Divide $100$ by $- 4$ .

$e = - 22 - - 25$

Step 6.1.4.2.1.4

Multiply $- 1$ by $- 25$ .

$e = - 22 + 25$

Step 6.1.4.2.2

Add $- 22$ and $25$ .

$e = 3$

Step 6.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 5)}^{2} + 3$ .

$- {(x - 5)}^{2} + 3$

Step 6.2

Set $y$ equal to the new right side.

$y = - {(x - 5)}^{2} + 3$

Step 7

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: Right $5$ Units

Step 8

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 $3$ Units

Step 9

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

Reflection about the x-axis: Reflected

Step 10

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

Reflection about the y-axis: None

Step 11

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: None

Step 12

Compare and list the transformations.

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

Horizontal Shift: Right $5$ Units

Vertical Shift: Up $3$ Units

Reflection about the x-axis: Reflected

Reflection about the y-axis: None

Vertical Compression or Stretch: None

Step 13