The problem asks to analyze the function $f(x) = x^4 - x^2$ and complete several parts. Here's a breakdown of the answers from the image:

1. Part (a) - Leading Coefficient Test:

   • The function $f(x) = x^4 - x^2$ is a quartic function (degree 4), and its leading term is $x^4$, which is positive.
   • According to the Leading Coefficient Test:
     • For even-degree functions with a positive leading coefficient, the graph rises to the left and rises to the right.
   • Correct answer: C (The graph of $f(x)$ rises left and rises right).

2. Part (b) - Finding x-intercepts:

   • To find the x-intercepts, set $f(x) = 0$: $x^4 - x^2 = 0$ Factor the expression: $x^2(x^2 - 1) = 0$ This gives two factors: $x^2 = 0 \quad \text{or} \quad x^2 - 1 = 0$ Solving: $x = 0 \quad \text{or} \quad x = \pm 1$ Thus, the x-intercepts are $x = 0, 1, -1$.
   • Correct answer: x = 0, 1, -1.

3. Part (c) - At which zeros does the graph cross the x-axis:

   • The graph crosses the x-axis at points where the multiplicity of the root is odd. In this case:
     • For $x = 0$, the root comes from $x^2 = 0$, which has even multiplicity, so it doesn't cross here (the graph touches the x-axis but doesn't cross).
     • For $x = \pm 1$, these come from linear factors, so the multiplicity is odd, and the graph crosses the x-axis at these points.
   • Correct answer: A (x = 1, -1).

4. Part (d) - At which zeros does the graph touch the x-axis and turn around:

   • The graph touches the x-axis and turns around at points where the multiplicity of the root is even.
   • For $x = 0$, the root $x^2 = 0$ has even multiplicity.
   • Correct answer: A (x = 0).

Let me know if you'd like further clarification or details on any of these points!

Related Questions:

1. What is the significance of the leading coefficient in determining end behavior?
2. How does the multiplicity of roots affect whether the graph crosses or touches the x-axis?
3. Why does a degree-4 polynomial have certain types of end behaviors?
4. Can a graph both touch and cross the x-axis at the same root?
5. How can factoring help in finding the x-intercepts of higher-degree polynomials?

Tip:

When solving polynomials, always check the multiplicity of roots to determine whether the graph crosses or merely touches the x-axis.