Algebra Examples

$y = \frac{1}{2} {(x + 1)}^{4} - 3$

Step 1

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

$y = x^{4}$

Step 2

Simplify $\frac{1}{2} \cdot {(x + 1)}^{4} - 3$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Use the Binomial Theorem.

$y = \frac{1}{2} \cdot (x^{4} + 4 x^{3} \cdot 1 + 6 x^{2} \cdot 1^{2} + 4 x \cdot 1^{3} + 1^{4}) - 3$

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

Multiply $4$ by $1$ .

$y = \frac{1}{2} \cdot (x^{4} + 4 x^{3} + 6 x^{2} \cdot 1^{2} + 4 x \cdot 1^{3} + 1^{4}) - 3$

Step 2.1.2.2

One to any power is one.

$y = \frac{1}{2} \cdot (x^{4} + 4 x^{3} + 6 x^{2} \cdot 1 + 4 x \cdot 1^{3} + 1^{4}) - 3$

Step 2.1.2.3

Multiply $6$ by $1$ .

$y = \frac{1}{2} \cdot (x^{4} + 4 x^{3} + 6 x^{2} + 4 x \cdot 1^{3} + 1^{4}) - 3$

Step 2.1.2.4

One to any power is one.

$y = \frac{1}{2} \cdot (x^{4} + 4 x^{3} + 6 x^{2} + 4 x \cdot 1 + 1^{4}) - 3$

Step 2.1.2.5

Multiply $4$ by $1$ .

$y = \frac{1}{2} \cdot (x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1^{4}) - 3$

Step 2.1.2.6

One to any power is one.

$y = \frac{1}{2} \cdot (x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1) - 3$

Step 2.1.3

Apply the distributive property.

$y = \frac{1}{2} x^{4} + \frac{1}{2} (4 x^{3}) + \frac{1}{2} (6 x^{2}) + \frac{1}{2} (4 x) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Combine $\frac{1}{2}$ and $x^{4}$ .

$y = \frac{x^{4}}{2} + \frac{1}{2} (4 x^{3}) + \frac{1}{2} (6 x^{2}) + \frac{1}{2} (4 x) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.2

Cancel the common factor of $2$ .

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Step 2.1.4.2.1

Factor $2$ out of $4 x^{3}$ .

$y = \frac{x^{4}}{2} + \frac{1}{2} (2 (2 x^{3})) + \frac{1}{2} (6 x^{2}) + \frac{1}{2} (4 x) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.2.2

Cancel the common factor.

$y = \frac{x^{4}}{2} + \frac{1}{2} (2 (2 x^{3})) + \frac{1}{2} (6 x^{2}) + \frac{1}{2} (4 x) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.2.3

Rewrite the expression.

$y = \frac{x^{4}}{2} + 2 x^{3} + \frac{1}{2} (6 x^{2}) + \frac{1}{2} (4 x) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.3

Cancel the common factor of $2$ .

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Step 2.1.4.3.1

Factor $2$ out of $6 x^{2}$ .

$y = \frac{x^{4}}{2} + 2 x^{3} + \frac{1}{2} (2 (3 x^{2})) + \frac{1}{2} (4 x) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.3.2

Cancel the common factor.

$y = \frac{x^{4}}{2} + 2 x^{3} + \frac{1}{2} (2 (3 x^{2})) + \frac{1}{2} (4 x) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.3.3

Rewrite the expression.

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + \frac{1}{2} (4 x) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.4

Cancel the common factor of $2$ .

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Step 2.1.4.4.1

Factor $2$ out of $4 x$ .

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + \frac{1}{2} (2 (2 x)) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.4.2

Cancel the common factor.

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + \frac{1}{2} (2 (2 x)) + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.4.3

Rewrite the expression.

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x + \frac{1}{2} \cdot 1 - 3$

Step 2.1.4.5

Multiply $\frac{1}{2}$ by $1$ .

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x + \frac{1}{2} - 3$

Step 2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x + \frac{1}{2} - 3 \cdot \frac{2}{2}$

Step 2.3

Combine $- 3$ and $\frac{2}{2}$ .

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x + \frac{1}{2} + \frac{- 3 \cdot 2}{2}$

Step 2.4

Combine the numerators over the common denominator.

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x + \frac{1 - 3 \cdot 2}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 3$ by $2$ .

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x + \frac{1 - 6}{2}$

Step 2.5.2

Subtract $6$ from $1$ .

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x + \frac{- 5}{2}$

Step 2.6

Move the negative in front of the fraction.

$y = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x - \frac{5}{2}$

Step 3

Assume that $y = x^{4}$ is $f (x) = x^{4}$ and $y = \frac{1}{2} \cdot {(x + 1)}^{4} - 3$ is $g (x) = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x - \frac{5}{2}$ .

$f (x) = x^{4}$

$g (x) = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x - \frac{5}{2}$

Step 4

The transformation being described is from $f (x) = x^{4}$ to $g (x) = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x - \frac{5}{2}$ .

$f (x) = x^{4} \to g (x) = \frac{x^{4}}{2} + 2 x^{3} + 3 x^{2} + 2 x - \frac{5}{2}$

Step 5

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 $1$ Units

Step 6

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: Down $3$ Units

Step 7

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

Reflection about the x-axis: None

Step 8

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

Reflection about the y-axis: None

Step 9

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 10

Compare and list the transformations.

Parent Function: $y = x^{4}$

Horizontal Shift: Left $1$ Units

Vertical Shift: Down $3$ Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: Compressed

Step 11