Algebra Examples

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

Step 1

Find the point at $x = - 2$ .

Tap for more steps...

Step 1.1

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

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

Step 1.2

Simplify the result.

Tap for more steps...

Step 1.2.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1

Multiply $2$ by $- 2$ .

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

Step 1.2.1.2

Add $- 4$ and $3$ .

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

Step 1.2.1.3

Raise $- 1$ to the power of $2$ .

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

Step 1.2.1.4

Multiply $- 2$ by $1$ .

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

Step 1.2.1.5

Multiply $2$ by $- 2$ .

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

Step 1.2.1.6

Add $- 4$ and $3$ .

$f (- 2) = - 2 - {(- 1)}^{2}$

Step 1.2.1.7

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Tap for more steps...

Step 1.2.1.7.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

Tap for more steps...

Step 1.2.1.7.1.1

Raise $- 1$ to the power of $1$ .

$f (- 2) = - 2 + (- 1) {(- 1)}^{2}$

Step 1.2.1.7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 2) = - 2 + {(- 1)}^{1 + 2}$

Step 1.2.1.7.2

Add $1$ and $2$ .

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

Step 1.2.1.8

Raise $- 1$ to the power of $3$ .

$f (- 2) = - 2 - 1$

Step 1.2.2

Subtract $1$ from $- 2$ .

$f (- 2) = - 3$

Step 1.2.3

The final answer is $- 3$ .

$- 3$

Step 1.3

Convert $- 3$ to decimal.

$y = - 3$

Step 2

Find the point at $x = - 1$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply $2$ by $- 1$ .

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

Step 2.2.1.2

Add $- 2$ and $3$ .

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

Step 2.2.1.3

One to any power is one.

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

Step 2.2.1.4

Multiply $- 1$ by $1$ .

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

Step 2.2.1.5

Multiply $2$ by $- 1$ .

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

Step 2.2.1.6

Add $- 2$ and $3$ .

$f (- 1) = - 1 - 1^{2}$

Step 2.2.1.7

One to any power is one.

$f (- 1) = - 1 - 1 \cdot 1$

Step 2.2.1.8

Multiply $- 1$ by $1$ .

$f (- 1) = - 1 - 1$

Step 2.2.2

Subtract $1$ from $- 1$ .

$f (- 1) = - 2$

Step 2.2.3

The final answer is $- 2$ .

$- 2$

Step 2.3

Convert $- 2$ to decimal.

$y = - 2$

Step 3

Find the point at $x = 0$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Multiply $2$ by $0$ .

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

Step 3.2.1.2

Add $0$ and $3$ .

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

Step 3.2.1.3

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

$f (0) = 0 \cdot 9 - {(2 (0) + 3)}^{2}$

Step 3.2.1.4

Multiply $0$ by $9$ .

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

Step 3.2.1.5

Multiply $2$ by $0$ .

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

Step 3.2.1.6

Add $0$ and $3$ .

$f (0) = 0 - 3^{2}$

Step 3.2.1.7

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

$f (0) = 0 - 1 \cdot 9$

Step 3.2.1.8

Multiply $- 1$ by $9$ .

$f (0) = 0 - 9$

Step 3.2.2

Subtract $9$ from $0$ .

$f (0) = - 9$

Step 3.2.3

The final answer is $- 9$ .

$- 9$

Step 3.3

Convert $- 9$ to decimal.

$y = - 9$

Step 4

Find the point at $x = 1$ .

Tap for more steps...

Step 4.1

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

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

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Multiply ${(2 (1) + 3)}^{2}$ by $1$ .

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

Step 4.2.1.2

Multiply $2$ by $1$ .

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

Step 4.2.1.3

Add $2$ and $3$ .

$f (1) = 5^{2} - {(2 (1) + 3)}^{2}$

Step 4.2.1.4

Raise $5$ to the power of $2$ .

$f (1) = 25 - {(2 (1) + 3)}^{2}$

Step 4.2.1.5

Multiply $2$ by $1$ .

$f (1) = 25 - {(2 + 3)}^{2}$

Step 4.2.1.6

Add $2$ and $3$ .

$f (1) = 25 - 5^{2}$

Step 4.2.1.7

Raise $5$ to the power of $2$ .

$f (1) = 25 - 1 \cdot 25$

Step 4.2.1.8

Multiply $- 1$ by $25$ .

$f (1) = 25 - 25$

Step 4.2.2

Subtract $25$ from $25$ .

$f (1) = 0$

Step 4.2.3

The final answer is $0$ .

$0$

Step 4.3

Convert $0$ to decimal.

$y = 0$

Step 5

Find the point at $x = 2$ .

Tap for more steps...

Step 5.1

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

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

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Multiply $2$ by $2$ .

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

Step 5.2.1.2

Add $4$ and $3$ .

$f (2) = 2 \cdot 7^{2} - {(2 (2) + 3)}^{2}$

Step 5.2.1.3

Raise $7$ to the power of $2$ .

$f (2) = 2 \cdot 49 - {(2 (2) + 3)}^{2}$

Step 5.2.1.4

Multiply $2$ by $49$ .

$f (2) = 98 - {(2 (2) + 3)}^{2}$

Step 5.2.1.5

Multiply $2$ by $2$ .

$f (2) = 98 - {(4 + 3)}^{2}$

Step 5.2.1.6

Add $4$ and $3$ .

$f (2) = 98 - 7^{2}$

Step 5.2.1.7

Raise $7$ to the power of $2$ .

$f (2) = 98 - 1 \cdot 49$

Step 5.2.1.8

Multiply $- 1$ by $49$ .

$f (2) = 98 - 49$

Step 5.2.2

Subtract $49$ from $98$ .

$f (2) = 49$

Step 5.2.3

The final answer is $49$ .

$49$

Step 5.3

Convert $49$ to decimal.

$y = 49$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & - 3 \\ - 1 & - 2 \\ 0 & - 9 \\ 1 & 0 \\ 2 & 49 \end{array}$

Step 7

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & - 3 \\ - 1 & - 2 \\ 0 & - 9 \\ 1 & 0 \\ 2 & 49 \end{array}$

Step 8