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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.

Use the graph of the quadratic function f to determine the solution: (a) Solve f(x) > 0. (b) Solve f(x) ≤ 0.

Learn how to solve quadratic inequalities using a graph. This problem demonstrates how to find intervals for f(x) > 0 and f(x) ≤ 0 based on a quadratic graph with intercepts at x = -5 and x = 5. Perfect for students in Grades 8-10.

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