Find the End Behavior f(x)=-5x(2x-5)^2

$f (x) = - 5 x {(2 x - 5)}^{2}$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

$- 5 x ((2 x - 5) (2 x - 5))$

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

Apply the distributive property.

$- 5 x (2 x (2 x - 5) - 5 (2 x - 5))$

Apply the distributive property.

$- 5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5))$

Apply the distributive property.

$- 5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 5 x (2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

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

Move $x$ .

$- 5 x (2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $x$ by $x$ .

$- 5 x (2 \cdot 2 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $2$ by $2$ .

$- 5 x (4 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $- 5$ by $2$ .

$- 5 x (4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5)$

Multiply $2$ by $- 5$ .

$- 5 x (4 x^{2} - 10 x - 10 x - 5 \cdot - 5)$

Multiply $- 5$ by $- 5$ .

$- 5 x (4 x^{2} - 10 x - 10 x + 25)$

Subtract $10 x$ from $- 10 x$ .

$- 5 x (4 x^{2} - 20 x + 25)$

Apply the distributive property.

$- 5 x (4 x^{2}) - 5 x (- 20 x) - 5 x \cdot 25$

Simplify.

Rewrite using the commutative property of multiplication.

$- 5 \cdot 4 x \cdot x^{2} - 5 x (- 20 x) - 5 x \cdot 25$

Rewrite using the commutative property of multiplication.

$- 5 \cdot 4 x \cdot x^{2} - 5 \cdot - 20 x \cdot x - 5 x \cdot 25$

Multiply $25$ by $- 5$ .

$- 5 \cdot 4 x \cdot x^{2} - 5 \cdot - 20 x \cdot x - 125 x$

Simplify each term.

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

Move $x^{2}$ .

$- 5 \cdot 4 (x^{2} x) - 5 \cdot - 20 x \cdot x - 125 x$

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

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

$- 5 \cdot 4 (x^{2} x^{1}) - 5 \cdot - 20 x \cdot x - 125 x$

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

$- 5 \cdot 4 x^{2 + 1} - 5 \cdot - 20 x \cdot x - 125 x$

Add $2$ and $1$ .

$- 5 \cdot 4 x^{3} - 5 \cdot - 20 x \cdot x - 125 x$

Multiply $- 5$ by $4$ .

$- 20 x^{3} - 5 \cdot - 20 x \cdot x - 125 x$

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

Move $x$ .

$- 20 x^{3} - 5 \cdot - 20 (x \cdot x) - 125 x$

Multiply $x$ by $x$ .

$- 20 x^{3} - 5 \cdot - 20 x^{2} - 125 x$

Multiply $- 5$ by $- 20$ .

$- 20 x^{3} + 100 x^{2} - 125 x$

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.

Rewrite ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

$- 5 x ((2 x - 5) (2 x - 5))$

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

Apply the distributive property.

$- 5 x (2 x (2 x - 5) - 5 (2 x - 5))$

Apply the distributive property.

$- 5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5))$

Apply the distributive property.

$- 5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 5 x (2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

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

Move $x$ .

$- 5 x (2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $x$ by $x$ .

$- 5 x (2 \cdot 2 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $2$ by $2$ .

$- 5 x (4 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $- 5$ by $2$ .

$- 5 x (4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5)$

Multiply $2$ by $- 5$ .

$- 5 x (4 x^{2} - 10 x - 10 x - 5 \cdot - 5)$

Multiply $- 5$ by $- 5$ .

$- 5 x (4 x^{2} - 10 x - 10 x + 25)$

Subtract $10 x$ from $- 10 x$ .

$- 5 x (4 x^{2} - 20 x + 25)$

Apply the distributive property.

$- 5 x (4 x^{2}) - 5 x (- 20 x) - 5 x \cdot 25$

Simplify.

Rewrite using the commutative property of multiplication.

$- 5 \cdot 4 x \cdot x^{2} - 5 x (- 20 x) - 5 x \cdot 25$

Rewrite using the commutative property of multiplication.

$- 5 \cdot 4 x \cdot x^{2} - 5 \cdot - 20 x \cdot x - 5 x \cdot 25$

Multiply $25$ by $- 5$ .

$- 5 \cdot 4 x \cdot x^{2} - 5 \cdot - 20 x \cdot x - 125 x$

Simplify each term.

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

Move $x^{2}$ .

$- 5 \cdot 4 (x^{2} x) - 5 \cdot - 20 x \cdot x - 125 x$

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

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

$- 5 \cdot 4 (x^{2} x^{1}) - 5 \cdot - 20 x \cdot x - 125 x$

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

$- 5 \cdot 4 x^{2 + 1} - 5 \cdot - 20 x \cdot x - 125 x$

Add $2$ and $1$ .

$- 5 \cdot 4 x^{3} - 5 \cdot - 20 x \cdot x - 125 x$

Multiply $- 5$ by $4$ .

$- 20 x^{3} - 5 \cdot - 20 x \cdot x - 125 x$

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

Move $x$ .

$- 20 x^{3} - 5 \cdot - 20 (x \cdot x) - 125 x$

Multiply $x$ by $x$ .

$- 20 x^{3} - 5 \cdot - 20 x^{2} - 125 x$

Multiply $- 5$ by $- 20$ .

$- 20 x^{3} + 100 x^{2} - 125 x$

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

$- 20 x^{3}$

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

$- 20$

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)=-5x(2x-5)^2

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