Find the End Behavior f(x)=3(x-7)(x+3)^2

$f (x) = 3 (x - 7) {(x + 3)}^{2}$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify by multiplying through.

Apply the distributive property.

$(3 x + 3 \cdot - 7) {(x + 3)}^{2}$

Simplify the expression.

Multiply $3$ by $- 7$ .

$(3 x - 21) {(x + 3)}^{2}$

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$(3 x - 21) ((x + 3) (x + 3))$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$(3 x - 21) (x (x + 3) + 3 (x + 3))$

Apply the distributive property.

$(3 x - 21) (x \cdot x + x \cdot 3 + 3 (x + 3))$

Apply the distributive property.

$(3 x - 21) (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(3 x - 21) (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3)$

Move $3$ to the left of $x$ .

$(3 x - 21) (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3)$

Multiply $3$ by $3$ .

$(3 x - 21) (x^{2} + 3 x + 3 x + 9)$

Add $3 x$ and $3 x$ .

$(3 x - 21) (x^{2} + 6 x + 9)$

Expand $(3 x - 21) (x^{2} + 6 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$3 x \cdot x^{2} + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$3 (x^{2} x) + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

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

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

$3 (x^{2} x^{1}) + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

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

$3 x^{2 + 1} + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Add $2$ and $1$ .

$3 x^{3} + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Rewrite using the commutative property of multiplication.

$3 x^{3} + 3 \cdot 6 x \cdot x + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

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

Move $x$ .

$3 x^{3} + 3 \cdot 6 (x \cdot x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Multiply $x$ by $x$ .

$3 x^{3} + 3 \cdot 6 x^{2} + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Multiply $3$ by $6$ .

$3 x^{3} + 18 x^{2} + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Multiply $9$ by $3$ .

$3 x^{3} + 18 x^{2} + 27 x - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Multiply $6$ by $- 21$ .

$3 x^{3} + 18 x^{2} + 27 x - 21 x^{2} - 126 x - 21 \cdot 9$

Multiply $- 21$ by $9$ .

$3 x^{3} + 18 x^{2} + 27 x - 21 x^{2} - 126 x - 189$

Simplify by adding terms.

Subtract $21 x^{2}$ from $18 x^{2}$ .

$3 x^{3} - 3 x^{2} + 27 x - 126 x - 189$

Subtract $126 x$ from $27 x$ .

$3 x^{3} - 3 x^{2} - 99 x - 189$

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.

$(3 x + 3 \cdot - 7) {(x + 3)}^{2}$

Simplify the expression.

Multiply $3$ by $- 7$ .

$(3 x - 21) {(x + 3)}^{2}$

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$(3 x - 21) ((x + 3) (x + 3))$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$(3 x - 21) (x (x + 3) + 3 (x + 3))$

Apply the distributive property.

$(3 x - 21) (x \cdot x + x \cdot 3 + 3 (x + 3))$

Apply the distributive property.

$(3 x - 21) (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(3 x - 21) (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3)$

Move $3$ to the left of $x$ .

$(3 x - 21) (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3)$

Multiply $3$ by $3$ .

$(3 x - 21) (x^{2} + 3 x + 3 x + 9)$

Add $3 x$ and $3 x$ .

$(3 x - 21) (x^{2} + 6 x + 9)$

Expand $(3 x - 21) (x^{2} + 6 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$3 x \cdot x^{2} + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$3 (x^{2} x) + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

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

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

$3 (x^{2} x^{1}) + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

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

$3 x^{2 + 1} + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Add $2$ and $1$ .

$3 x^{3} + 3 x (6 x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Rewrite using the commutative property of multiplication.

$3 x^{3} + 3 \cdot 6 x \cdot x + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

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

Move $x$ .

$3 x^{3} + 3 \cdot 6 (x \cdot x) + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Multiply $x$ by $x$ .

$3 x^{3} + 3 \cdot 6 x^{2} + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Multiply $3$ by $6$ .

$3 x^{3} + 18 x^{2} + 3 x \cdot 9 - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Multiply $9$ by $3$ .

$3 x^{3} + 18 x^{2} + 27 x - 21 x^{2} - 21 (6 x) - 21 \cdot 9$

Multiply $6$ by $- 21$ .

$3 x^{3} + 18 x^{2} + 27 x - 21 x^{2} - 126 x - 21 \cdot 9$

Multiply $- 21$ by $9$ .

$3 x^{3} + 18 x^{2} + 27 x - 21 x^{2} - 126 x - 189$

Simplify by adding terms.

Subtract $21 x^{2}$ from $18 x^{2}$ .

$3 x^{3} - 3 x^{2} + 27 x - 126 x - 189$

Subtract $126 x$ from $27 x$ .

$3 x^{3} - 3 x^{2} - 99 x - 189$

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

$3 x^{3}$

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

$3$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

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.

Falls to the left and rises to the right

Step 7

Find the End Behavior f(x)=3(x-7)(x+3)^2

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