Let's go through each part of this problem step-by-step.

Given Polynomial:

$f(x) = x^4 + 8x^3 - 9x$

Step-by-Step Solution

Part (a): Finding the Zeros and Their Multiplicities

1. Factor the polynomial:
   $f(x) = x(x^3 + 8x^2 - 9)$
   Factor the cubic $x^3 + 8x^2 - 9 = 0$ further if possible to find all roots.

   By solving this equation, you find:

   • One real zero, $x = 0$, with a multiplicity of 1.

2. Listing the zero and multiplicity:

   • Zero $x = 0$, multiplicity $m = 1$.

Part (b): Determine Touch or Cross at $x$-intercepts

Since $x = 0$ has an odd multiplicity (1), the graph will cross the x-axis at this point.

Part (c): Finding the $y$-intercept and Function Values at Specific Points

1. $y$-intercept:
   The $y$-intercept occurs when $x = 0$.
   Substitute $x = 0$ into $f(x)$:
   $f(0) = 0$
   So, the $y$-intercept is at $(0, 0)$.

2. Finding $f(-1)$:
   Substitute $x = -1$:
   $f(-1) = (-1)^4 + 8(-1)^3 - 9(-1) = 1 - 8 + 9 = 2$

3. Finding $f(2)$:
   Substitute $x = 2$:
   $f(2) = (2)^4 + 8(2)^3 - 9(2) = 16 + 64 - 18 = 62$

Part (d): Determine the End Behavior

To determine end behavior, examine the highest degree term, $x^4$. Since it is positive, as $x \to \infty$, $f(x) \to \infty$, and as $x \to -\infty$, $f(x) \to \infty$. This tells us the graph rises to infinity in both directions.

Part (e): Selecting the Correct Graph

The correct graph should:

• Cross the x-axis at $x = 0$.
• Have end behavior going up in both directions.
• Pass through the point $(0, 0)$ for the y-intercept.

Based on these details, you should be able to match it to one of the provided graphs.

Would you like further clarification on any of these parts, or assistance with another specific question?

Here are five additional questions related to this problem:

1. How would the end behavior change if the leading term had a negative coefficient?
2. What would happen if the zero had an even multiplicity?
3. How would you find complex roots for higher-degree polynomials if necessary?
4. How does the multiplicity of a zero affect the shape of the graph at that zero?
5. How can the derivative of $f(x)$ help in understanding its local maxima and minima?

Tip: Always check the multiplicity of zeros to determine whether the graph will touch or cross at each intercept.