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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.

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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.

Consider the polynomial f(x) = x^4 + 8x^3 - 9x.

Learn how to analyze and graph the polynomial f(x) = x^4 + 8x^3 - 9x. This problem includes finding zeros and their multiplicities, determining end behavior, and identifying the correct graph. Suitable for high school students in Grades 10-12.

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