Algebra Examples

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

Step 1

Write $\frac{3}{2} (x^{3} - 2)$ as an equation.

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

Step 2

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

$y = x^{3}$

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Combine $\frac{3}{2}$ and $x^{3}$ .

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

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Factor $2$ out of $- 2$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Multiply $3$ by $- 1$ .

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

Step 4

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

$f (x) = x^{3}$

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

Step 5

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

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

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.

In this case, $h = 0$ which means that the graph is not shifted to the left or right.

Horizontal Shift: None

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: Down $3$ Units

Step 8

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

Reflection about the x-axis: None

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: Stretched

Step 11

Compare and list the transformations.

Parent Function: $y = x^{3}$

Horizontal Shift: None

Vertical Shift: Down $3$ Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: Stretched

Step 12