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Let's solve the given quadratic function \( f(x) = 9x^2 - 25 \).

### Step 1: Finding the Vertex
The general form of a quadratic function is \( f(x) = ax^2 + bx + c \). Here, we have:
- \( a = 9 \)
- \( b = 0 \)
- \( c = -25 \)

For a parabola of the form \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is found using the formula:
\[
x_{\text{vertex}} = \frac{-b}{2a}
\]
Since \( b = 0 \), the x-coordinate of the vertex is:
\[
x_{\text{vertex}} = \frac{0}{2(9)} = 0
\]
To find the y-coordinate of the vertex, substitute \( x = 0 \) into the function:
\[
f(0) = 9(0)^2 - 25 = -25
\]
Thus, the vertex is \( (0, -25) \).

### Step 2: Finding the x-Intercepts
To find the x-intercepts, we set \( f(x) = 0 \) and solve for \( x \):
\[
9x^2 - 25 = 0
\]
\[
9x^2 = 25
\]
\[
x^2 = \frac{25}{9}
\]
\[
x = \pm \frac{5}{3}
\]
The x-intercepts are \( x = \frac{5}{3} \) and \( x = -\frac{5}{3} \). The largest x-intercept is \( \frac{5}{3} \).

### Step 3: Finding the y-Intercept
The y-intercept occurs when \( x = 0 \). We've already calculated \( f(0) = -25 \), so the y-intercept is \( y = -25 \).

### Summary of Answers:
- Vertex: \( (0, -25) \)
- Largest x-intercept: \( \frac{5}{3} \)
- y-intercept: \( -25 \)

Would you like further details on any of these steps or have any questions? Here are five related questions for further exploration:

1. How do we determine the axis of symmetry from the vertex?
2. What is the significance of the sign of the coefficient \( a \) in a quadratic function?
3. How do you find the discriminant of a quadratic equation, and what does it tell us about the intercepts?
4. Can a quadratic function have more than two x-intercepts?
5. How does the standard form of a quadratic differ from its vertex form?

**Tip:** The vertex form of a quadratic function \( f(x) = a(x - h)^2 + k \) makes it easier to identify the vertex directly without needing to calculate it from the standard form.

Consider the quadratic function f(x) = 9x^2 - 25. Its vertex is (___ , ___). The x value of its largest x-intercept is x = ___. The y value of the y-intercept is y = ___.

Learn how to find the vertex, x-intercepts, and y-intercept of the quadratic function f(x) = 9x^2 - 25. This problem demonstrates key algebraic concepts related to parabolas and their properties. Suitable for students in Grades 9-10.

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