Graph y=(2x+3)(x-1)(x-4)

$y = (2 x + 3) (x - 1) (x - 4)$

Step 1

Find the point at $x = 0$ .

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

$f (0) = 2 {(0)}^{3} - 7 {(0)}^{2} - 7 \cdot 0 + 12$

Simplify the result.

Simplify each term.

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

$f (0) = 2 \cdot 0 - 7 {(0)}^{2} - 7 \cdot 0 + 12$

Multiply $2$ by $0$ .

$f (0) = 0 - 7 {(0)}^{2} - 7 \cdot 0 + 12$

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

$f (0) = 0 - 7 \cdot 0 - 7 \cdot 0 + 12$

Multiply $- 7$ by $0$ .

$f (0) = 0 + 0 - 7 \cdot 0 + 12$

Multiply $- 7$ by $0$ .

$f (0) = 0 + 0 + 0 + 12$

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 0 + 12$

Add $0$ and $0$ .

$f (0) = 0 + 12$

Add $0$ and $12$ .

$f (0) = 12$

The final answer is $12$ .

$12$

Convert $12$ to decimal.

$y = 12$

Step 2

Find the point at $x = 1$ .

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

$f (1) = 2 {(1)}^{3} - 7 {(1)}^{2} - 7 \cdot 1 + 12$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 2 \cdot 1 - 7 {(1)}^{2} - 7 \cdot 1 + 12$

Multiply $2$ by $1$ .

$f (1) = 2 - 7 {(1)}^{2} - 7 \cdot 1 + 12$

One to any power is one.

$f (1) = 2 - 7 \cdot 1 - 7 \cdot 1 + 12$

Multiply $- 7$ by $1$ .

$f (1) = 2 - 7 - 7 \cdot 1 + 12$

Multiply $- 7$ by $1$ .

$f (1) = 2 - 7 - 7 + 12$

Simplify by adding and subtracting.

Subtract $7$ from $2$ .

$f (1) = - 5 - 7 + 12$

Subtract $7$ from $- 5$ .

$f (1) = - 12 + 12$

Add $- 12$ and $12$ .

$f (1) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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

$f (2) = 2 {(2)}^{3} - 7 {(2)}^{2} - 7 \cdot 2 + 12$

Simplify the result.

Simplify each term.

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

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

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

$f (2) = 2 {(2)}^{3} - 7 {(2)}^{2} - 7 \cdot 2 + 12$

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

$f (2) = 2^{1 + 3} - 7 {(2)}^{2} - 7 \cdot 2 + 12$

Add $1$ and $3$ .

$f (2) = 2^{4} - 7 {(2)}^{2} - 7 \cdot 2 + 12$

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

$f (2) = 16 - 7 {(2)}^{2} - 7 \cdot 2 + 12$

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

$f (2) = 16 - 7 \cdot 4 - 7 \cdot 2 + 12$

Multiply $- 7$ by $4$ .

$f (2) = 16 - 28 - 7 \cdot 2 + 12$

Multiply $- 7$ by $2$ .

$f (2) = 16 - 28 - 14 + 12$

Simplify by adding and subtracting.

Subtract $28$ from $16$ .

$f (2) = - 12 - 14 + 12$

Subtract $14$ from $- 12$ .

$f (2) = - 26 + 12$

Add $- 26$ and $12$ .

$f (2) = - 14$

The final answer is $- 14$ .

$- 14$

Convert $- 14$ to decimal.

$y = - 14$

Step 4

Find the point at $x = 3$ .

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

$f (3) = 2 {(3)}^{3} - 7 {(3)}^{2} - 7 \cdot 3 + 12$

Simplify the result.

Simplify each term.

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

$f (3) = 2 \cdot 27 - 7 {(3)}^{2} - 7 \cdot 3 + 12$

Multiply $2$ by $27$ .

$f (3) = 54 - 7 {(3)}^{2} - 7 \cdot 3 + 12$

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

$f (3) = 54 - 7 \cdot 9 - 7 \cdot 3 + 12$

Multiply $- 7$ by $9$ .

$f (3) = 54 - 63 - 7 \cdot 3 + 12$

Multiply $- 7$ by $3$ .

$f (3) = 54 - 63 - 21 + 12$

Simplify by adding and subtracting.

Subtract $63$ from $54$ .

$f (3) = - 9 - 21 + 12$

Subtract $21$ from $- 9$ .

$f (3) = - 30 + 12$

Add $- 30$ and $12$ .

$f (3) = - 18$

The final answer is $- 18$ .

$- 18$

Convert $- 18$ to decimal.

$y = - 18$

Step 5

Find the point at $x = 4$ .

Replace the variable $x$ with $4$ in the expression.

$f (4) = 2 {(4)}^{3} - 7 {(4)}^{2} - 7 \cdot 4 + 12$

Simplify the result.

Simplify each term.

Raise $4$ to the power of $3$ .

$f (4) = 2 \cdot 64 - 7 {(4)}^{2} - 7 \cdot 4 + 12$

Multiply $2$ by $64$ .

$f (4) = 128 - 7 {(4)}^{2} - 7 \cdot 4 + 12$

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

$f (4) = 128 - 7 \cdot 16 - 7 \cdot 4 + 12$

Multiply $- 7$ by $16$ .

$f (4) = 128 - 112 - 7 \cdot 4 + 12$

Multiply $- 7$ by $4$ .

$f (4) = 128 - 112 - 28 + 12$

Simplify by adding and subtracting.

Subtract $112$ from $128$ .

$f (4) = 16 - 28 + 12$

Subtract $28$ from $16$ .

$f (4) = - 12 + 12$

Add $- 12$ and $12$ .

$f (4) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 6

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

$\begin{array}{c} x & y \\ 0 & 12 \\ 1 & 0 \\ 2 & - 14 \\ 3 & - 18 \\ 4 & 0 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & 12 \\ 1 & 0 \\ 2 & - 14 \\ 3 & - 18 \\ 4 & 0 \end{array}$

Step 8

Graph y=(2x+3)(x-1)(x-4)

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