Precalculus Examples

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

Step 1

Find the point at $x = 0$ .

Tap for more steps...

Step 1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{2} - \frac{{(0)}^{3}}{4} - 4 + 0$

Step 1.2

Simplify the result.

Tap for more steps...

Step 1.2.1

Remove parentheses.

$f (0) = {(0)}^{2} - \frac{{(0)}^{3}}{4} - 4 + 0$

Step 1.2.2

Simplify each term.

Tap for more steps...

Step 1.2.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - \frac{{(0)}^{3}}{4} - 4 + 0$

Step 1.2.2.2

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - \frac{0}{4} - 4 + 0$

Step 1.2.2.3

Divide $0$ by $4$ .

$f (0) = 0 - 0 - 4 + 0$

Step 1.2.2.4

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0 - 4 + 0$

Step 1.2.3

Simplify by adding and subtracting.

Tap for more steps...

Step 1.2.3.1

Add $0$ and $0$ .

$f (0) = 0 - 4 + 0$

Step 1.2.3.2

Subtract $4$ from $0$ .

$f (0) = - 4 + 0$

Step 1.2.3.3

Add $- 4$ and $0$ .

$f (0) = - 4$

Step 1.2.4

The final answer is $- 4$ .

$- 4$

Step 1.3

Convert $- 4$ to decimal.

$y = - 4$

Step 2

Find the point at $x = 2$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = {(2)}^{2} - \frac{{(2)}^{3}}{4} - 4 + 2$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$f (2) = {(2)}^{2} - \frac{{(2)}^{3}}{4} - 4 + 2$

Step 2.2.2

Simplify each term.

Tap for more steps...

Step 2.2.2.1

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

$f (2) = 4 - \frac{{(2)}^{3}}{4} - 4 + 2$

Step 2.2.2.2

Raise $2$ to the power of $3$ .

$f (2) = 4 - \frac{8}{4} - 4 + 2$

Step 2.2.2.3

Divide $8$ by $4$ .

$f (2) = 4 - 1 \cdot 2 - 4 + 2$

Step 2.2.2.4

Multiply $- 1$ by $2$ .

$f (2) = 4 - 2 - 4 + 2$

Step 2.2.3

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.3.1

Subtract $2$ from $4$ .

$f (2) = 2 - 4 + 2$

Step 2.2.3.2

Subtract $4$ from $2$ .

$f (2) = - 2 + 2$

Step 2.2.3.3

Add $- 2$ and $2$ .

$f (2) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 1$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $1$ in the expression.

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Remove parentheses.

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

Step 3.2.2

Simplify each term.

Tap for more steps...

Step 3.2.2.1

One to any power is one.

$f (1) = 1 - \frac{{(1)}^{3}}{4} - 4 + 1$

Step 3.2.2.2

One to any power is one.

$f (1) = 1 - \frac{1}{4} - 4 + 1$

Step 3.2.3

Find the common denominator.

Tap for more steps...

Step 3.2.3.1

Write $1$ as a fraction with denominator $1$ .

$f (1) = \frac{1}{1} - \frac{1}{4} - 4 + 1$

Step 3.2.3.2

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$f (1) = \frac{1}{1} \cdot \frac{4}{4} - \frac{1}{4} - 4 + 1$

Step 3.2.3.3

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$f (1) = \frac{4}{4} - \frac{1}{4} - 4 + 1$

Step 3.2.3.4

Write $- 4$ as a fraction with denominator $1$ .

$f (1) = \frac{4}{4} - \frac{1}{4} + \frac{- 4}{1} + 1$

Step 3.2.3.5

Multiply $\frac{- 4}{1}$ by $\frac{4}{4}$ .

$f (1) = \frac{4}{4} - \frac{1}{4} + \frac{- 4}{1} \cdot \frac{4}{4} + 1$

Step 3.2.3.6

Multiply $\frac{- 4}{1}$ by $\frac{4}{4}$ .

$f (1) = \frac{4}{4} - \frac{1}{4} + \frac{- 4 \cdot 4}{4} + 1$

Step 3.2.3.7

Write $1$ as a fraction with denominator $1$ .

$f (1) = \frac{4}{4} - \frac{1}{4} + \frac{- 4 \cdot 4}{4} + \frac{1}{1}$

Step 3.2.3.8

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$f (1) = \frac{4}{4} - \frac{1}{4} + \frac{- 4 \cdot 4}{4} + \frac{1}{1} \cdot \frac{4}{4}$

Step 3.2.3.9

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$f (1) = \frac{4}{4} - \frac{1}{4} + \frac{- 4 \cdot 4}{4} + \frac{4}{4}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (1) = \frac{4 - 1 - 4 \cdot 4 + 4}{4}$

Step 3.2.5

Simplify the expression.

Tap for more steps...

Step 3.2.5.1

Multiply $- 4$ by $4$ .

$f (1) = \frac{4 - 1 - 16 + 4}{4}$

Step 3.2.5.2

Subtract $1$ from $4$ .

$f (1) = \frac{3 - 16 + 4}{4}$

Step 3.2.5.3

Subtract $16$ from $3$ .

$f (1) = \frac{- 13 + 4}{4}$

Step 3.2.5.4

Add $- 13$ and $4$ .

$f (1) = \frac{- 9}{4}$

Step 3.2.5.5

Move the negative in front of the fraction.

$f (1) = - \frac{9}{4}$

Step 3.2.6

The final answer is $- \frac{9}{4}$ .

$- \frac{9}{4}$

Step 3.3

Convert $- \frac{9}{4}$ to decimal.

$y = - 2.25$

Step 4

Find the point at $x = 3$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = {(3)}^{2} - \frac{{(3)}^{3}}{4} - 4 + 3$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Remove parentheses.

$f (3) = {(3)}^{2} - \frac{{(3)}^{3}}{4} - 4 + 3$

Step 4.2.2

Simplify each term.

Tap for more steps...

Step 4.2.2.1

Raise $3$ to the power of $2$ .

$f (3) = 9 - \frac{{(3)}^{3}}{4} - 4 + 3$

Step 4.2.2.2

Raise $3$ to the power of $3$ .

$f (3) = 9 - \frac{27}{4} - 4 + 3$

Step 4.2.3

Find the common denominator.

Tap for more steps...

Step 4.2.3.1

Write $9$ as a fraction with denominator $1$ .

$f (3) = \frac{9}{1} - \frac{27}{4} - 4 + 3$

Step 4.2.3.2

Multiply $\frac{9}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9}{1} \cdot \frac{4}{4} - \frac{27}{4} - 4 + 3$

Step 4.2.3.3

Multiply $\frac{9}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9 \cdot 4}{4} - \frac{27}{4} - 4 + 3$

Step 4.2.3.4

Write $- 4$ as a fraction with denominator $1$ .

$f (3) = \frac{9 \cdot 4}{4} - \frac{27}{4} + \frac{- 4}{1} + 3$

Step 4.2.3.5

Multiply $\frac{- 4}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9 \cdot 4}{4} - \frac{27}{4} + \frac{- 4}{1} \cdot \frac{4}{4} + 3$

Step 4.2.3.6

Multiply $\frac{- 4}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9 \cdot 4}{4} - \frac{27}{4} + \frac{- 4 \cdot 4}{4} + 3$

Step 4.2.3.7

Write $3$ as a fraction with denominator $1$ .

$f (3) = \frac{9 \cdot 4}{4} - \frac{27}{4} + \frac{- 4 \cdot 4}{4} + \frac{3}{1}$

Step 4.2.3.8

Multiply $\frac{3}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9 \cdot 4}{4} - \frac{27}{4} + \frac{- 4 \cdot 4}{4} + \frac{3}{1} \cdot \frac{4}{4}$

Step 4.2.3.9

Multiply $\frac{3}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9 \cdot 4}{4} - \frac{27}{4} + \frac{- 4 \cdot 4}{4} + \frac{3 \cdot 4}{4}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (3) = \frac{9 \cdot 4 - 27 - 4 \cdot 4 + 3 \cdot 4}{4}$

Step 4.2.5

Simplify each term.

Tap for more steps...

Step 4.2.5.1

Multiply $9$ by $4$ .

$f (3) = \frac{36 - 27 - 4 \cdot 4 + 3 \cdot 4}{4}$

Step 4.2.5.2

Multiply $- 4$ by $4$ .

$f (3) = \frac{36 - 27 - 16 + 3 \cdot 4}{4}$

Step 4.2.5.3

Multiply $3$ by $4$ .

$f (3) = \frac{36 - 27 - 16 + 12}{4}$

Step 4.2.6

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.6.1

Subtract $27$ from $36$ .

$f (3) = \frac{9 - 16 + 12}{4}$

Step 4.2.6.2

Subtract $16$ from $9$ .

$f (3) = \frac{- 7 + 12}{4}$

Step 4.2.6.3

Add $- 7$ and $12$ .

$f (3) = \frac{5}{4}$

Step 4.2.7

The final answer is $\frac{5}{4}$ .

$\frac{5}{4}$

Step 4.3

Convert $\frac{5}{4}$ to decimal.

$y = 1.25$

Step 5

The cubic function can be graphed using the function behavior and the points.

$\begin{array}{c} x & y \\ 0 & - 4 \\ 1 & - 2.25 \\ 2 & 0 \\ 3 & 1.25 \\ 4 & 0 \end{array}$

Step 6

The cubic function can be graphed using the function behavior and the selected points.

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ 0 & - 4 \\ 1 & - 2.25 \\ 2 & 0 \\ 3 & 1.25 \\ 4 & 0 \end{array}$

Step 7