Algebra Examples

$y = (2 x + 5) {(x - 1)}^{2}$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

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

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 $- 2$ .

$f (- 2) = - 4 {((- 2) - 1)}^{2} + 5 {((- 2) - 1)}^{2}$

Step 1.2.1.2

Subtract $1$ from $- 2$ .

$f (- 2) = - 4 {(- 3)}^{2} + 5 {((- 2) - 1)}^{2}$

Step 1.2.1.3

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

$f (- 2) = - 4 \cdot 9 + 5 {((- 2) - 1)}^{2}$

Step 1.2.1.4

Multiply $- 4$ by $9$ .

$f (- 2) = - 36 + 5 {((- 2) - 1)}^{2}$

Step 1.2.1.5

Subtract $1$ from $- 2$ .

$f (- 2) = - 36 + 5 {(- 3)}^{2}$

Step 1.2.1.6

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

$f (- 2) = - 36 + 5 \cdot 9$

Step 1.2.1.7

Multiply $5$ by $9$ .

$f (- 2) = - 36 + 45$

Step 1.2.2

Add $- 36$ and $45$ .

$f (- 2) = 9$

Step 1.2.3

The final answer is $9$ .

$9$

Step 1.3

Convert $9$ to decimal.

$y = 9$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

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

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 $- 1$ .

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

Step 2.2.1.2

Subtract $1$ from $- 1$ .

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

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

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Step 2.2.1.3.1.1

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

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

Step 2.2.1.3.1.2

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

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

Step 2.2.1.3.2

Add $1$ and $2$ .

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

Step 2.2.1.4

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

$f (- 1) = - 8 + 5 {((- 1) - 1)}^{2}$

Step 2.2.1.5

Subtract $1$ from $- 1$ .

$f (- 1) = - 8 + 5 {(- 2)}^{2}$

Step 2.2.1.6

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

$f (- 1) = - 8 + 5 \cdot 4$

Step 2.2.1.7

Multiply $5$ by $4$ .

$f (- 1) = - 8 + 20$

Step 2.2.2

Add $- 8$ and $20$ .

$f (- 1) = 12$

Step 2.2.3

The final answer is $12$ .

$12$

Step 2.3

Convert $12$ to decimal.

$y = 12$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = 2 (0) {((0) - 1)}^{2} + 5 {((0) - 1)}^{2}$

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 $0$ .

$f (0) = 0 {((0) - 1)}^{2} + 5 {((0) - 1)}^{2}$

Step 3.2.1.2

Subtract $1$ from $0$ .

$f (0) = 0 {(- 1)}^{2} + 5 {((0) - 1)}^{2}$

Step 3.2.1.3

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

$f (0) = 0 \cdot 1 + 5 {((0) - 1)}^{2}$

Step 3.2.1.4

Multiply $0$ by $1$ .

$f (0) = 0 + 5 {((0) - 1)}^{2}$

Step 3.2.1.5

Subtract $1$ from $0$ .

$f (0) = 0 + 5 {(- 1)}^{2}$

Step 3.2.1.6

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

$f (0) = 0 + 5 \cdot 1$

Step 3.2.1.7

Multiply $5$ by $1$ .

$f (0) = 0 + 5$

Step 3.2.2

Add $0$ and $5$ .

$f (0) = 5$

Step 3.2.3

The final answer is $5$ .

$5$

Step 3.3

Convert $5$ to decimal.

$y = 5$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

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

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 $1$ .

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

Step 4.2.1.2

Subtract $1$ from $1$ .

$f (1) = 2 \cdot 0^{2} + 5 {((1) - 1)}^{2}$

Step 4.2.1.3

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

$f (1) = 2 \cdot 0 + 5 {((1) - 1)}^{2}$

Step 4.2.1.4

Multiply $2$ by $0$ .

$f (1) = 0 + 5 {((1) - 1)}^{2}$

Step 4.2.1.5

Subtract $1$ from $1$ .

$f (1) = 0 + 5 \cdot 0^{2}$

Step 4.2.1.6

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

$f (1) = 0 + 5 \cdot 0$

Step 4.2.1.7

Multiply $5$ by $0$ .

$f (1) = 0 + 0$

Step 4.2.2

Add $0$ and $0$ .

$f (1) = 0$

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 $x = 2$ .

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Step 5.1

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

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

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 $2$ .

$f (2) = 4 {((2) - 1)}^{2} + 5 {((2) - 1)}^{2}$

Step 5.2.1.2

Subtract $1$ from $2$ .

$f (2) = 4 \cdot 1^{2} + 5 {((2) - 1)}^{2}$

Step 5.2.1.3

One to any power is one.

$f (2) = 4 \cdot 1 + 5 {((2) - 1)}^{2}$

Step 5.2.1.4

Multiply $4$ by $1$ .

$f (2) = 4 + 5 {((2) - 1)}^{2}$

Step 5.2.1.5

Subtract $1$ from $2$ .

$f (2) = 4 + 5 \cdot 1^{2}$

Step 5.2.1.6

One to any power is one.

$f (2) = 4 + 5 \cdot 1$

Step 5.2.1.7

Multiply $5$ by $1$ .

$f (2) = 4 + 5$

Step 5.2.2

Add $4$ and $5$ .

$f (2) = 9$

Step 5.2.3

The final answer is $9$ .

$9$

Step 5.3

Convert $9$ to decimal.

$y = 9$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 9 \\ - 1 & 12 \\ 0 & 5 \\ 1 & 0 \\ 2 & 9 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & 9 \\ - 1 & 12 \\ 0 & 5 \\ 1 & 0 \\ 2 & 9 \end{array}$

Step 8