In the given graph, you are asked to solve the quadratic function $f(x)$ for two conditions:

• (a) $f(x) > 0$
• (b) $f(x) \leq 0$

Understanding the graph:

The graph shows a parabola that opens upwards with two x-intercepts at $x = -5$ and $x = 5$. These are the roots of the quadratic equation, where $f(x) = 0$. The vertex of the parabola is located at $x = 0$, which corresponds to the minimum point on the graph, and from the graph, the function is negative between $-5$ and $5$, and positive outside of that interval.

(a) Solving $f(x) > 0$:

For $f(x) > 0$, we are looking for the intervals where the function is above the x-axis. From the graph, the function is positive when:

• $x < -5$
• $x > 5$

Thus, the solution to $f(x) > 0$ in interval notation is: $(-\infty, -5) \cup (5, \infty)$

(b) Solving $f(x) \leq 0$:

For $f(x) \leq 0$, the function is either on or below the x-axis. From the graph, this occurs between the x-intercepts, including the points where $f(x) = 0$, which are $x = -5$ and $x = 5$.

Thus, the solution to $f(x) \leq 0$ in interval notation is: $[-5, 5]$

Would you like more details or have any other questions?

Here are 5 questions that expand on this topic:

1. How do we derive the equation of a quadratic function from a graph?
2. What is the significance of the vertex in a parabola?
3. How can we determine whether a quadratic function opens upward or downward?
4. How would the solutions change if the parabola opened downward?
5. How does the discriminant help in understanding the roots of a quadratic function?

Tip: When working with inequalities involving quadratic functions, always observe the intervals where the function is positive or negative relative to the x-axis.