Algebra Examples

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

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $2$ by $- 1$ .

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

Step 1.2.1.2

Subtract $1$ from $- 1$ .

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

Step 1.2.1.3

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

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Step 1.2.1.3.1

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

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Step 1.2.1.3.1.1

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

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

Step 1.2.1.3.1.2

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

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

Step 1.2.1.3.2

Add $1$ and $2$ .

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

Step 1.2.1.4

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

$f (- 1) = - 8 + 3 {((- 1) - 1)}^{2}$

Step 1.2.1.5

Subtract $1$ from $- 1$ .

$f (- 1) = - 8 + 3 {(- 2)}^{2}$

Step 1.2.1.6

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

$f (- 1) = - 8 + 3 \cdot 4$

Step 1.2.1.7

Multiply $3$ by $4$ .

$f (- 1) = - 8 + 12$

Step 1.2.2

Add $- 8$ and $12$ .

$f (- 1) = 4$

Step 1.2.3

The final answer is $4$ .

$4$

Step 1.3

Convert $4$ to decimal.

$y = 4$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

$f (0) = 2 (0) {((0) - 1)}^{2} + 3 {((0) - 1)}^{2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $2$ by $0$ .

$f (0) = 0 {((0) - 1)}^{2} + 3 {((0) - 1)}^{2}$

Step 2.2.1.2

Subtract $1$ from $0$ .

$f (0) = 0 {(- 1)}^{2} + 3 {((0) - 1)}^{2}$

Step 2.2.1.3

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

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

Step 2.2.1.4

Multiply $0$ by $1$ .

$f (0) = 0 + 3 {((0) - 1)}^{2}$

Step 2.2.1.5

Subtract $1$ from $0$ .

$f (0) = 0 + 3 {(- 1)}^{2}$

Step 2.2.1.6

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

$f (0) = 0 + 3 \cdot 1$

Step 2.2.1.7

Multiply $3$ by $1$ .

$f (0) = 0 + 3$

Step 2.2.2

Add $0$ and $3$ .

$f (0) = 3$

Step 2.2.3

The final answer is $3$ .

$3$

Step 2.3

Convert $3$ to decimal.

$y = 3$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $2$ by $1$ .

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

Step 3.2.1.2

Subtract $1$ from $1$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

Multiply $2$ by $0$ .

$f (1) = 0 + 3 {((1) - 1)}^{2}$

Step 3.2.1.5

Subtract $1$ from $1$ .

$f (1) = 0 + 3 \cdot 0^{2}$

Step 3.2.1.6

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

$f (1) = 0 + 3 \cdot 0$

Step 3.2.1.7

Multiply $3$ by $0$ .

$f (1) = 0 + 0$

Step 3.2.2

Add $0$ and $0$ .

$f (1) = 0$

Step 3.2.3

The final answer is $0$ .

$0$

Step 3.3

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $2$ by $2$ .

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

Step 4.2.1.2

Subtract $1$ from $2$ .

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

Step 4.2.1.3

One to any power is one.

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

Step 4.2.1.4

Multiply $4$ by $1$ .

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

Step 4.2.1.5

Subtract $1$ from $2$ .

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

Step 4.2.1.6

One to any power is one.

$f (2) = 4 + 3 \cdot 1$

Step 4.2.1.7

Multiply $3$ by $1$ .

$f (2) = 4 + 3$

Step 4.2.2

Add $4$ and $3$ .

$f (2) = 7$

Step 4.2.3

The final answer is $7$ .

$7$

Step 4.3

Convert $7$ to decimal.

$y = 7$

Step 5

Find the point at $x = 3$ .

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $2$ by $3$ .

$f (3) = 6 {((3) - 1)}^{2} + 3 {((3) - 1)}^{2}$

Step 5.2.1.2

Subtract $1$ from $3$ .

$f (3) = 6 \cdot 2^{2} + 3 {((3) - 1)}^{2}$

Step 5.2.1.3

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

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

Step 5.2.1.4

Multiply $6$ by $4$ .

$f (3) = 24 + 3 {((3) - 1)}^{2}$

Step 5.2.1.5

Subtract $1$ from $3$ .

$f (3) = 24 + 3 \cdot 2^{2}$

Step 5.2.1.6

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

$f (3) = 24 + 3 \cdot 4$

Step 5.2.1.7

Multiply $3$ by $4$ .

$f (3) = 24 + 12$

Step 5.2.2

Add $24$ and $12$ .

$f (3) = 36$

Step 5.2.3

The final answer is $36$ .

$36$

Step 5.3

Convert $36$ to decimal.

$y = 36$

Step 6

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

$\begin{array}{c} x & y \\ - 1 & 4 \\ 0 & 3 \\ 1 & 0 \\ 2 & 7 \\ 3 & 36 \end{array}$

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

$\begin{array}{c} x & y \\ - 1 & 4 \\ 0 & 3 \\ 1 & 0 \\ 2 & 7 \\ 3 & 36 \end{array}$

Step 8