Question

consider the function (p(x) = (x - 1)^2(x - 3))

the (y)-intercept is the point

the (x)-intercept(s) is/are the point(s)

as (x \to infty), (y \to)

as (x \to -infty), (y \to)

question 11

given (f(x) = -9(x + 1)^4(x + 2)^4(2x - 1)(x + 4)^2)

list the zeroes (ex. 4, 6, -2):

degree:

end behavior:

as (x \to -infty), (f(x) \to)
as (x \to +infty), (f(x) \to)

Explanation:

⚡ Using what you learned: Graphing Polynomial Functions

Step 1: Analyze the first function $P(x)$

We are given:

$$ P(x) = (x-1)^2(x-3) $$

Find the $y$-intercept

Set \( x = 0 \):

$$ P(0) = (0-1)^2(0-3) = (1)(-3) = -3 $$

The $y$-intercept is the point \( (0, -3) \).

Find the $x$-intercept(s)

Set \( P(x) = 0 \):

$$ (x-1)^2(x-3) = 0 $$

This gives \( x = 1 \) and \( x = 3 \).
The $x$-intercepts are the points \( (1, 0), (3, 0) \).

Determine end behavior

The leading term of \( P(x) \) is found by multiplying the highest degree terms of each factor:

$$ x^2 \cdot x = x^3 $$

Since the degree is odd (\( 3 \)) and the leading coefficient is positive (\( 1 \)):

• As \( x \to \infty \), \( y \to \infty \)
• As \( x \to -\infty \), \( y \to -\infty \)

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Step 2: Analyze the second function $f(x)$

We are given:

$$ f(x) = -9(x+1)^4(x+2)^4(2x-1)(x+4)^2 $$

List the zeroes

Set each factor to zero:

• \( x + 1 = 0 \implies x = -1 \)
• \( x + 2 = 0 \implies x = -2 \)
• \( 2x - 1 = 0 \implies x = \frac{1}{2} \) (or \( 0.5 \))
• \( x + 4 = 0 \implies x = -4 \)

The zeroes are: \( -1, -2, 0.5, -4 \)

Find the degree

Sum the exponents of the variable factors:

$$ \text{Degree} = 4 + 4 + 1 + 2 = 11 $$

Determine end behavior

The leading term is:

$$ -9 \cdot (x)^4 \cdot (x)^4 \cdot (2x)^1 \cdot (x)^2 = -18x^{11} $$

Since the degree is odd (\( 11 \)) and the leading coefficient is negative (\( -18 \)):

• As \( x \to -\infty \), \( f(x) \to \infty \)
• As \( x \to \infty \), \( f(x) \to -\infty \)

Answer:

For the first question:

• The $y$-intercept is the point: (0, -3)
• The $x$-intercept(s) is/are the point(s): (1, 0), (3, 0)
• As \( x \to \infty \), \( y \to \) \infty (or oo)
• As \( x \to -\infty \), \( y \to \) -\infty (or -oo)

For Question 11:

• List the zeroes: -1, -2, 0.5, -4
• Degree: 11
• End behavior:
• As \( x \to -\infty \), \( f(x) \to \) \infty (or oo)
• As \( x \to \infty \), \( f(x) \to \) -\infty (or -oo)