Pre-Algebra Examples

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

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

$f (0) = - e {(0)}^{3} + 7 e {(0)}^{2} - 7 e \cdot 0$

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

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

$f (0) = - e \cdot 0 + 7 e {(0)}^{2} - 7 e \cdot 0$

Step 1.2.1.2

Multiply $- e \cdot 0$ .

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Step 1.2.1.2.1

Multiply $0$ by $- 1$ .

$f (0) = 0 e + 7 e {(0)}^{2} - 7 e \cdot 0$

Step 1.2.1.2.2

Multiply $0$ by $e$ .

$f (0) = 0 + 7 e {(0)}^{2} - 7 e \cdot 0$

Step 1.2.1.3

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

$f (0) = 0 + 7 e \cdot 0 - 7 e \cdot 0$

Step 1.2.1.4

Multiply $7 e \cdot 0$ .

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Step 1.2.1.4.1

Multiply $0$ by $7$ .

$f (0) = 0 + 0 e - 7 e \cdot 0$

Step 1.2.1.4.2

Multiply $0$ by $e$ .

$f (0) = 0 + 0 - 7 e \cdot 0$

Step 1.2.1.5

Multiply $- 7 e \cdot 0$ .

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Step 1.2.1.5.1

Multiply $0$ by $- 7$ .

$f (0) = 0 + 0 + 0 e$

Step 1.2.1.5.2

Multiply $0$ by $e$ .

$f (0) = 0 + 0 + 0$

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 1.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = - 2$ .

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Step 2.1

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

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

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

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

$f (- 2) = - e \cdot - 8 + 7 e {(- 2)}^{2} - 7 e \cdot - 2$

Step 2.2.1.2

Multiply $- 8$ by $- 1$ .

$f (- 2) = 8 e + 7 e {(- 2)}^{2} - 7 e \cdot - 2$

Step 2.2.1.3

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

$f (- 2) = 8 e + 7 e \cdot 4 - 7 e \cdot - 2$

Step 2.2.1.4

Multiply $4$ by $7$ .

$f (- 2) = 8 e + 28 e - 7 e \cdot - 2$

Step 2.2.1.5

Multiply $- 2$ by $- 7$ .

$f (- 2) = 8 e + 28 e + 14 e$

Step 2.2.2

Simplify by adding terms.

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Step 2.2.2.1

Add $8 e$ and $28 e$ .

$f (- 2) = 36 e + 14 e$

Step 2.2.2.2

Add $36 e$ and $14 e$ .

$f (- 2) = 50 e$

Step 2.2.3

The final answer is $50 e$ .

$50 e$

Step 2.3

Convert $50 e$ to decimal.

$y = 135.91409142$

Step 3

Find the point at $x = - 1$ .

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Step 3.1

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

$f (- 1) = - e {(- 1)}^{3} + 7 e {(- 1)}^{2} - 7 e \cdot - 1$

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 $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 3.2.1.1.1

Move ${(- 1)}^{3}$ .

$f (- 1) = {(- 1)}^{3} \cdot (- 1 e) + 7 e {(- 1)}^{2} - 7 e \cdot - 1$

Step 3.2.1.1.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

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Step 3.2.1.1.2.1

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

$f (- 1) = {(- 1)}^{3} \cdot ((- 1) e) + 7 e {(- 1)}^{2} - 7 e \cdot - 1$

Step 3.2.1.1.2.2

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

$f (- 1) = {(- 1)}^{3 + 1} e + 7 e {(- 1)}^{2} - 7 e \cdot - 1$

Step 3.2.1.1.3

Add $3$ and $1$ .

$f (- 1) = {(- 1)}^{4} e + 7 e {(- 1)}^{2} - 7 e \cdot - 1$

Step 3.2.1.2

Raise $- 1$ to the power of $4$ .

$f (- 1) = 1 e + 7 e {(- 1)}^{2} - 7 e \cdot - 1$

Step 3.2.1.3

Multiply $e$ by $1$ .

$f (- 1) = e + 7 e {(- 1)}^{2} - 7 e \cdot - 1$

Step 3.2.1.4

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

$f (- 1) = e + 7 e \cdot 1 - 7 e \cdot - 1$

Step 3.2.1.5

Multiply $7$ by $1$ .

$f (- 1) = e + 7 e - 7 e \cdot - 1$

Step 3.2.1.6

Multiply $- 1$ by $- 7$ .

$f (- 1) = e + 7 e + 7 e$

Step 3.2.2

Simplify by adding terms.

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Step 3.2.2.1

Add $e$ and $7 e$ .

$f (- 1) = 8 e + 7 e$

Step 3.2.2.2

Add $8 e$ and $7 e$ .

$f (- 1) = 15 e$

Step 3.2.3

The final answer is $15 e$ .

$15 e$

Step 3.3

Convert $15 e$ to decimal.

$y = 40.77422742$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = - e {(1)}^{3} + 7 e {(1)}^{2} - 7 e \cdot 1$

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

One to any power is one.

$f (1) = - e \cdot 1 + 7 e {(1)}^{2} - 7 e \cdot 1$

Step 4.2.1.2

Multiply $- 1$ by $1$ .

$f (1) = - e + 7 e {(1)}^{2} - 7 e \cdot 1$

Step 4.2.1.3

One to any power is one.

$f (1) = - e + 7 e \cdot 1 - 7 e \cdot 1$

Step 4.2.1.4

Multiply $7$ by $1$ .

$f (1) = - e + 7 e - 7 e \cdot 1$

Step 4.2.1.5

Multiply $- 7$ by $1$ .

$f (1) = - e + 7 e - 7 e$

Step 4.2.2

Simplify by adding terms.

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Step 4.2.2.1

Add $- e$ and $7 e$ .

$f (1) = 6 e - 7 e$

Step 4.2.2.2

Subtract $7 e$ from $6 e$ .

$f (1) = - e$

Step 4.2.3

The final answer is $- e$ .

$- e$

Step 4.3

Convert $- e$ to decimal.

$y = - 2.71828182$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

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

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

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

$f (2) = - e \cdot 8 + 7 e {(2)}^{2} - 7 e \cdot 2$

Step 5.2.1.2

Multiply $8$ by $- 1$ .

$f (2) = - 8 e + 7 e {(2)}^{2} - 7 e \cdot 2$

Step 5.2.1.3

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

$f (2) = - 8 e + 7 e \cdot 4 - 7 e \cdot 2$

Step 5.2.1.4

Multiply $4$ by $7$ .

$f (2) = - 8 e + 28 e - 7 e \cdot 2$

Step 5.2.1.5

Multiply $2$ by $- 7$ .

$f (2) = - 8 e + 28 e - 14 e$

Step 5.2.2

Simplify by adding terms.

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Step 5.2.2.1

Add $- 8 e$ and $28 e$ .

$f (2) = 20 e - 14 e$

Step 5.2.2.2

Subtract $14 e$ from $20 e$ .

$f (2) = 6 e$

Step 5.2.3

The final answer is $6 e$ .

$6 e$

Step 5.3

Convert $6 e$ to decimal.

$y = 16.30969097$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 135.914 \\ - 1 & 40.774 \\ 0 & 0 \\ 1 & - 2.718 \\ 2 & 16.31 \end{array}$

Step 7

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

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ - 2 & 135.914 \\ - 1 & 40.774 \\ 0 & 0 \\ 1 & - 2.718 \\ 2 & 16.31 \end{array}$

Step 8