Algebra Examples

$f (x) = \frac{1}{16} (x + 2) (x - 2) (x - 4)$

Step 1

Find the point at $x = 0$ .

Tap for more steps...

Step 1.1

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

$f (0) = \frac{{(0)}^{3}}{16} - \frac{{(0)}^{2}}{4} - \frac{0}{4} + 1$

Step 1.2

Simplify the result.

Tap for more steps...

Step 1.2.1

Find the common denominator.

Tap for more steps...

Step 1.2.1.1

Multiply $\frac{{(0)}^{2}}{4}$ by $\frac{4}{4}$ .

$f (0) = \frac{{(0)}^{3}}{16} - (\frac{{(0)}^{2}}{4} \cdot \frac{4}{4}) - \frac{0}{4} + 1$

Step 1.2.1.2

Multiply $\frac{{(0)}^{2}}{4}$ by $\frac{4}{4}$ .

$f (0) = \frac{{(0)}^{3}}{16} - \frac{{(0)}^{2} \cdot 4}{4 \cdot 4} - \frac{0}{4} + 1$

Step 1.2.1.3

Multiply $\frac{0}{4}$ by $\frac{4}{4}$ .

$f (0) = \frac{{(0)}^{3}}{16} - \frac{{(0)}^{2} \cdot 4}{4 \cdot 4} - (\frac{0}{4} \cdot \frac{4}{4}) + 1$

Step 1.2.1.4

Multiply $\frac{0}{4}$ by $\frac{4}{4}$ .

$f (0) = \frac{{(0)}^{3}}{16} - \frac{{(0)}^{2} \cdot 4}{4 \cdot 4} - \frac{0 \cdot 4}{4 \cdot 4} + 1$

Step 1.2.1.5

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

$f (0) = \frac{{(0)}^{3}}{16} - \frac{{(0)}^{2} \cdot 4}{4 \cdot 4} - \frac{0 \cdot 4}{4 \cdot 4} + \frac{1}{1}$

Step 1.2.1.6

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

$f (0) = \frac{{(0)}^{3}}{16} - \frac{{(0)}^{2} \cdot 4}{4 \cdot 4} - \frac{0 \cdot 4}{4 \cdot 4} + \frac{1}{1} \cdot \frac{16}{16}$

Step 1.2.1.7

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

$f (0) = \frac{{(0)}^{3}}{16} - \frac{{(0)}^{2} \cdot 4}{4 \cdot 4} - \frac{0 \cdot 4}{4 \cdot 4} + \frac{16}{16}$

Step 1.2.1.8

Multiply $4$ by $4$ .

$f (0) = \frac{{(0)}^{3}}{16} - \frac{{(0)}^{2} \cdot 4}{16} - \frac{0 \cdot 4}{4 \cdot 4} + \frac{16}{16}$

Step 1.2.1.9

Multiply $4$ by $4$ .

$f (0) = \frac{{(0)}^{3}}{16} - \frac{{(0)}^{2} \cdot 4}{16} - \frac{0 \cdot 4}{16} + \frac{16}{16}$

Step 1.2.2

Combine the numerators over the common denominator.

$f (0) = \frac{{(0)}^{3} - {(0)}^{2} \cdot 4 + 0 \cdot 4 + 16}{16}$

Step 1.2.3

Simplify each term.

Tap for more steps...

Step 1.2.3.1

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

$f (0) = \frac{{(0)}^{3} + 0 \cdot 4 + 0 \cdot 4 + 16}{16}$

Step 1.2.3.2

Multiply $- 0 \cdot 4$ .

Tap for more steps...

Step 1.2.3.2.1

Multiply $- 1$ by $0$ .

$f (0) = \frac{{(0)}^{3} + 0 \cdot 4 + 0 \cdot 4 + 16}{16}$

Step 1.2.3.2.2

Multiply $0$ by $4$ .

$f (0) = \frac{{(0)}^{3} + 0 + 0 \cdot 4 + 16}{16}$

Step 1.2.3.3

Multiply $0$ by $4$ .

$f (0) = \frac{{(0)}^{3} + 0 + 0 + 16}{16}$

Step 1.2.4

Simplify the expression.

Tap for more steps...

Step 1.2.4.1

Add ${(0)}^{3}$ and $0$ .

$f (0) = \frac{{(0)}^{3} + 0 + 16}{16}$

Step 1.2.4.2

Add ${(0)}^{3}$ and $0$ .

$f (0) = \frac{{(0)}^{3} + 16}{16}$

Step 1.2.4.3

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

$f (0) = \frac{0 + 16}{16}$

Step 1.2.4.4

Add $0$ and $16$ .

$f (0) = \frac{16}{16}$

Step 1.2.4.5

Divide $16$ by $16$ .

$f (0) = 1$

Step 1.2.5

The final answer is $1$ .

$1$

Step 1.3

Convert $1$ to decimal.

$y = 1$

Step 2

Find the point at $x = 2$ .

Tap for more steps...

Step 2.1

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

$f (2) = \frac{{(2)}^{3}}{16} - \frac{{(2)}^{2}}{4} - \frac{2}{4} + 1$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Combine the numerators over the common denominator.

$f (2) = 1 + \frac{{(2)}^{3}}{16} + \frac{- {(2)}^{2} - 2}{4}$

Step 2.2.2

Simplify each term.

Tap for more steps...

Step 2.2.2.1

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

$f (2) = 1 + \frac{{(2)}^{3}}{16} + \frac{- 1 \cdot 4 - 2}{4}$

Step 2.2.2.2

Multiply $- 1$ by $4$ .

$f (2) = 1 + \frac{{(2)}^{3}}{16} + \frac{- 4 - 2}{4}$

Step 2.2.3

Subtract $2$ from $- 4$ .

$f (2) = 1 + \frac{{(2)}^{3}}{16} + \frac{- 6}{4}$

Step 2.2.4

Simplify each term.

Tap for more steps...

Step 2.2.4.1

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

$f (2) = 1 + \frac{8}{16} + \frac{- 6}{4}$

Step 2.2.4.2

Cancel the common factor of $8$ and $16$ .

Tap for more steps...

Step 2.2.4.2.1

Factor $8$ out of $8$ .

$f (2) = 1 + \frac{8 (1)}{16} + \frac{- 6}{4}$

Step 2.2.4.2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.4.2.2.1

Factor $8$ out of $16$ .

$f (2) = 1 + \frac{8 \cdot 1}{8 \cdot 2} + \frac{- 6}{4}$

Step 2.2.4.2.2.2

Cancel the common factor.

$f (2) = 1 + \frac{8 \cdot 1}{8 \cdot 2} + \frac{- 6}{4}$

Step 2.2.4.2.2.3

Rewrite the expression.

$f (2) = 1 + \frac{1}{2} + \frac{- 6}{4}$

Step 2.2.4.3

Cancel the common factor of $- 6$ and $4$ .

Tap for more steps...

Step 2.2.4.3.1

Factor $2$ out of $- 6$ .

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

Step 2.2.4.3.2

Cancel the common factors.

Tap for more steps...

Step 2.2.4.3.2.1

Factor $2$ out of $4$ .

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

Step 2.2.4.3.2.2

Cancel the common factor.

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

Step 2.2.4.3.2.3

Rewrite the expression.

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

Step 2.2.4.4

Move the negative in front of the fraction.

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

Step 2.2.5

Combine fractions.

Tap for more steps...

Step 2.2.5.1

Combine the numerators over the common denominator.

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

Step 2.2.5.2

Simplify the expression.

Tap for more steps...

Step 2.2.5.2.1

Subtract $3$ from $1$ .

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

Step 2.2.5.2.2

Divide $- 2$ by $2$ .

$f (2) = 1 - 1$

Step 2.2.5.2.3

Subtract $1$ from $1$ .

$f (2) = 0$

Step 2.2.6

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.

$f (1) = \frac{{(1)}^{3}}{16} - \frac{{(1)}^{2}}{4} - \frac{1}{4} + 1$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Combine the numerators over the common denominator.

$f (1) = 1 + \frac{{(1)}^{3}}{16} + \frac{- {(1)}^{2} - 1}{4}$

Step 3.2.2

Simplify each term.

Tap for more steps...

Step 3.2.2.1

One to any power is one.

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

Step 3.2.2.2

Multiply $- 1$ by $1$ .

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

Step 3.2.3

Subtract $1$ from $- 1$ .

$f (1) = 1 + \frac{{(1)}^{3}}{16} + \frac{- 2}{4}$

Step 3.2.4

Simplify each term.

Tap for more steps...

Step 3.2.4.1

One to any power is one.

$f (1) = 1 + \frac{1}{16} + \frac{- 2}{4}$

Step 3.2.4.2

Cancel the common factor of $- 2$ and $4$ .

Tap for more steps...

Step 3.2.4.2.1

Factor $2$ out of $- 2$ .

$f (1) = 1 + \frac{1}{16} + \frac{2 (- 1)}{4}$

Step 3.2.4.2.2

Cancel the common factors.

Tap for more steps...

Step 3.2.4.2.2.1

Factor $2$ out of $4$ .

$f (1) = 1 + \frac{1}{16} + \frac{2 \cdot - 1}{2 \cdot 2}$

Step 3.2.4.2.2.2

Cancel the common factor.

$f (1) = 1 + \frac{1}{16} + \frac{2 \cdot - 1}{2 \cdot 2}$

Step 3.2.4.2.2.3

Rewrite the expression.

$f (1) = 1 + \frac{1}{16} + \frac{- 1}{2}$

Step 3.2.4.3

Move the negative in front of the fraction.

$f (1) = 1 + \frac{1}{16} - \frac{1}{2}$

Step 3.2.5

Find the common denominator.

Tap for more steps...

Step 3.2.5.1

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

$f (1) = \frac{1}{1} + \frac{1}{16} - \frac{1}{2}$

Step 3.2.5.2

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

$f (1) = \frac{1}{1} \cdot \frac{16}{16} + \frac{1}{16} - \frac{1}{2}$

Step 3.2.5.3

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

$f (1) = \frac{16}{16} + \frac{1}{16} - \frac{1}{2}$

Step 3.2.5.4

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

$f (1) = \frac{16}{16} + \frac{1}{16} - (\frac{1}{2} \cdot \frac{8}{8})$

Step 3.2.5.5

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

$f (1) = \frac{16}{16} + \frac{1}{16} - \frac{8}{2 \cdot 8}$

Step 3.2.5.6

Multiply $2$ by $8$ .

$f (1) = \frac{16}{16} + \frac{1}{16} - \frac{8}{16}$

Step 3.2.6

Combine the numerators over the common denominator.

$f (1) = \frac{16 + 1 - 8}{16}$

Step 3.2.7

Simplify by adding and subtracting.

Tap for more steps...

Step 3.2.7.1

Add $16$ and $1$ .

$f (1) = \frac{17 - 8}{16}$

Step 3.2.7.2

Subtract $8$ from $17$ .

$f (1) = \frac{9}{16}$

Step 3.2.8

The final answer is $\frac{9}{16}$ .

$\frac{9}{16}$

Step 3.3

Convert $\frac{9}{16}$ to decimal.

$y = 0.5625$

Step 4

Find the point at $x = 3$ .

Tap for more steps...

Step 4.1

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

$f (3) = \frac{{(3)}^{3}}{16} - \frac{{(3)}^{2}}{4} - \frac{3}{4} + 1$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Combine the numerators over the common denominator.

$f (3) = 1 + \frac{{(3)}^{3}}{16} + \frac{- {(3)}^{2} - 3}{4}$

Step 4.2.2

Simplify each term.

Tap for more steps...

Step 4.2.2.1

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

$f (3) = 1 + \frac{{(3)}^{3}}{16} + \frac{- 1 \cdot 9 - 3}{4}$

Step 4.2.2.2

Multiply $- 1$ by $9$ .

$f (3) = 1 + \frac{{(3)}^{3}}{16} + \frac{- 9 - 3}{4}$

Step 4.2.3

Subtract $3$ from $- 9$ .

$f (3) = 1 + \frac{{(3)}^{3}}{16} + \frac{- 12}{4}$

Step 4.2.4

Simplify each term.

Tap for more steps...

Step 4.2.4.1

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

$f (3) = 1 + \frac{27}{16} + \frac{- 12}{4}$

Step 4.2.4.2

Divide $- 12$ by $4$ .

$f (3) = 1 + \frac{27}{16} - 3$

Step 4.2.5

Find the common denominator.

Tap for more steps...

Step 4.2.5.1

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

$f (3) = \frac{1}{1} + \frac{27}{16} - 3$

Step 4.2.5.2

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

$f (3) = \frac{1}{1} \cdot \frac{16}{16} + \frac{27}{16} - 3$

Step 4.2.5.3

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

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

Step 4.2.5.4

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

$f (3) = \frac{16}{16} + \frac{27}{16} + \frac{- 3}{1}$

Step 4.2.5.5

Multiply $\frac{- 3}{1}$ by $\frac{16}{16}$ .

$f (3) = \frac{16}{16} + \frac{27}{16} + \frac{- 3}{1} \cdot \frac{16}{16}$

Step 4.2.5.6

Multiply $\frac{- 3}{1}$ by $\frac{16}{16}$ .

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

Step 4.2.6

Combine the numerators over the common denominator.

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

Step 4.2.7

Simplify the expression.

Tap for more steps...

Step 4.2.7.1

Multiply $- 3$ by $16$ .

$f (3) = \frac{16 + 27 - 48}{16}$

Step 4.2.7.2

Add $16$ and $27$ .

$f (3) = \frac{43 - 48}{16}$

Step 4.2.7.3

Subtract $48$ from $43$ .

$f (3) = \frac{- 5}{16}$

Step 4.2.7.4

Move the negative in front of the fraction.

$f (3) = - \frac{5}{16}$

Step 4.2.8

The final answer is $- \frac{5}{16}$ .

$- \frac{5}{16}$

Step 4.3

Convert $- \frac{5}{16}$ to decimal.

$y = - 0.3125$

Step 5

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

$\begin{array}{c} x & y \\ 0 & 1 \\ 1 & 0.562 \\ 2 & 0 \\ 3 & - 0.312 \\ 4 & 0 \end{array}$

Step 6

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ 0 & 1 \\ 1 & 0.562 \\ 2 & 0 \\ 3 & - 0.312 \\ 4 & 0 \end{array}$

Step 7