Graph p(x)=2x^3+5x^2+x-2

$p (x) = 2 x^{3} + 5 x^{2} + x - 2$

Step 1

Find the point at $x = - 2$ .

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

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

Simplify the result.

Remove parentheses.

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

Simplify each term.

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

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

Multiply $2$ by $- 8$ .

$f (- 2) = - 16 + 5 {(- 2)}^{2} - 2 - 2$

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

$f (- 2) = - 16 + 5 \cdot 4 - 2 - 2$

Multiply $5$ by $4$ .

$f (- 2) = - 16 + 20 - 2 - 2$

Simplify by adding and subtracting.

Add $- 16$ and $20$ .

$f (- 2) = 4 - 2 - 2$

Subtract $2$ from $4$ .

$f (- 2) = 2 - 2$

Subtract $2$ from $2$ .

$f (- 2) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = 0$ .

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

$f (0) = 2 {(0)}^{3} + 5 {(0)}^{2} + 0 - 2$

Simplify the result.

Remove parentheses.

$f (0) = 2 {(0)}^{3} + 5 {(0)}^{2} + 0 - 2$

Simplify each term.

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

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

Multiply $2$ by $0$ .

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

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

$f (0) = 0 + 5 \cdot 0 + 0 - 2$

Multiply $5$ by $0$ .

$f (0) = 0 + 0 + 0 - 2$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 + 0 - 2$

Add $0$ and $0$ .

$f (0) = 0 - 2$

Subtract $2$ from $0$ .

$f (0) = - 2$

The final answer is $- 2$ .

$- 2$

Convert $- 2$ to decimal.

$y = - 2$

Step 3

Find the point at $x = 1$ .

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

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

Simplify the result.

Remove parentheses.

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

Simplify each term.

One to any power is one.

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

Multiply $2$ by $1$ .

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

One to any power is one.

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

Multiply $5$ by $1$ .

$f (1) = 2 + 5 + 1 - 2$

Simplify by adding and subtracting.

Add $2$ and $5$ .

$f (1) = 7 + 1 - 2$

Add $7$ and $1$ .

$f (1) = 8 - 2$

Subtract $2$ from $8$ .

$f (1) = 6$

The final answer is $6$ .

$6$

Convert $6$ to decimal.

$y = 6$

Step 4

Find the point at $x = 2$ .

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

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

Simplify the result.

Remove parentheses.

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

Simplify each term.

Multiply $2$ by ${(2)}^{3}$ by adding the exponents.

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

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

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

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

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

Add $1$ and $3$ .

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

Raise $2$ to the power of $4$ .

$f (2) = 16 + 5 {(2)}^{2} + 2 - 2$

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

$f (2) = 16 + 5 \cdot 4 + 2 - 2$

Multiply $5$ by $4$ .

$f (2) = 16 + 20 + 2 - 2$

Simplify by adding and subtracting.

Add $16$ and $20$ .

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

Add $36$ and $2$ .

$f (2) = 38 - 2$

Subtract $2$ from $38$ .

$f (2) = 36$

The final answer is $36$ .

$36$

Convert $36$ to decimal.

$y = 36$

Step 5

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

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 0 \\ 0 & - 2 \\ 1 & 6 \\ 2 & 36 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 0 \\ 0 & - 2 \\ 1 & 6 \\ 2 & 36 \end{array}$

Step 7

Graph p(x)=2x^3+5x^2+x-2

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