Find the End Behavior f(x)=-(x+1)(x-1)(x-3)

$f (x) = - (x + 1) (x - 1) (x - 3)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify by multiplying through.

Apply the distributive property.

$(- x - 1 \cdot 1) (x - 1) (x - 3)$

Multiply $- 1$ by $1$ .

$(- x - 1) (x - 1) (x - 3)$

Expand $(- x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$(- x (x - 1) - 1 (x - 1)) (x - 3)$

Apply the distributive property.

$(- x \cdot x - x \cdot - 1 - 1 (x - 1)) (x - 3)$

Apply the distributive property.

$(- x \cdot x - x \cdot - 1 - 1 x - 1 \cdot - 1) (x - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(- (x \cdot x) - x \cdot - 1 - 1 x - 1 \cdot - 1) (x - 3)$

Multiply $x$ by $x$ .

$(- x^{2} - x \cdot - 1 - 1 x - 1 \cdot - 1) (x - 3)$

Multiply $- x \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$(- x^{2} + 1 x - 1 x - 1 \cdot - 1) (x - 3)$

Multiply $x$ by $1$ .

$(- x^{2} + x - 1 x - 1 \cdot - 1) (x - 3)$

Rewrite $- 1 x$ as $- x$ .

$(- x^{2} + x - x - 1 \cdot - 1) (x - 3)$

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $x$ .

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

Add $- x^{2}$ and $0$ .

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

Expand $(- x^{2} + 1) (x - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$- x^{2} x - x^{2} \cdot - 3 + 1 (x - 3)$

Apply the distributive property.

$- x^{2} x - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$- (x \cdot x^{2}) - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

Multiply $x$ by $x^{2}$ .

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

$- (x^{1} x^{2}) - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

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

$- x^{1 + 2} - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

Add $1$ and $2$ .

$- x^{3} - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

Multiply $- 3$ by $- 1$ .

$- x^{3} + 3 x^{2} + 1 x + 1 \cdot - 3$

Multiply $x$ by $1$ .

$- x^{3} + 3 x^{2} + x + 1 \cdot - 3$

Multiply $- 3$ by $1$ .

$- x^{3} + 3 x^{2} + x - 3$

The largest exponent is the degree of the polynomial.

$3$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Simplify by multiplying through.

Apply the distributive property.

$(- x - 1 \cdot 1) (x - 1) (x - 3)$

Multiply $- 1$ by $1$ .

$(- x - 1) (x - 1) (x - 3)$

Expand $(- x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$(- x (x - 1) - 1 (x - 1)) (x - 3)$

Apply the distributive property.

$(- x \cdot x - x \cdot - 1 - 1 (x - 1)) (x - 3)$

Apply the distributive property.

$(- x \cdot x - x \cdot - 1 - 1 x - 1 \cdot - 1) (x - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(- (x \cdot x) - x \cdot - 1 - 1 x - 1 \cdot - 1) (x - 3)$

Multiply $x$ by $x$ .

$(- x^{2} - x \cdot - 1 - 1 x - 1 \cdot - 1) (x - 3)$

Multiply $- x \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$(- x^{2} + 1 x - 1 x - 1 \cdot - 1) (x - 3)$

Multiply $x$ by $1$ .

$(- x^{2} + x - 1 x - 1 \cdot - 1) (x - 3)$

Rewrite $- 1 x$ as $- x$ .

$(- x^{2} + x - x - 1 \cdot - 1) (x - 3)$

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $x$ .

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

Add $- x^{2}$ and $0$ .

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

Expand $(- x^{2} + 1) (x - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$- x^{2} x - x^{2} \cdot - 3 + 1 (x - 3)$

Apply the distributive property.

$- x^{2} x - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$- (x \cdot x^{2}) - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

Multiply $x$ by $x^{2}$ .

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

$- (x^{1} x^{2}) - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

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

$- x^{1 + 2} - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

Add $1$ and $2$ .

$- x^{3} - x^{2} \cdot - 3 + 1 x + 1 \cdot - 3$

Multiply $- 3$ by $- 1$ .

$- x^{3} + 3 x^{2} + 1 x + 1 \cdot - 3$

Multiply $x$ by $1$ .

$- x^{3} + 3 x^{2} + x + 1 \cdot - 3$

Multiply $- 3$ by $1$ .

$- x^{3} + 3 x^{2} + x - 3$

The leading term in a polynomial is the term with the highest degree.

$- x^{3}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 1$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Rises to the left and falls to the right

Step 7

Find the End Behavior f(x)=-(x+1)(x-1)(x-3)

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