https://file.aiquickdraw.com/mathbot/file/1728348074425gkmpi3li.jpg

The graph shows a parabola, and the question asks for two things:

**a)** To determine the function \( f(x) \) (i.e., write the equation).

**b)** To determine \( f(-4) \).

### Step a: Determine \( f(x) \)
The equation of a parabola is generally given in the form:

\[
f(x) = a(x - h)^2 + k
\]

Where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) is a constant that determines the width and direction (concave up or down).

Looking at the graph:
- The vertex is at \( (0, 16) \), so \( h = 0 \) and \( k = 16 \).
- The parabola opens downward (concave down), so \( a \) must be negative.

Next, to find \( a \), we can use another point on the parabola. From the graph, it looks like the parabola passes through the point \( (2, 12) \).

Using this point in the vertex form of the equation:
\[
f(2) = a(2 - 0)^2 + 16
\]
\[
12 = a(4) + 16
\]
\[
12 - 16 = 4a
\]
\[
a = -1
\]

Thus, the equation of the parabola is:
\[
f(x) = -(x - 0)^2 + 16 \quad \text{or simply} \quad f(x) = -x^2 + 16
\]

### Step b: Determine \( f(-4) \)
Now that we know the equation \( f(x) = -x^2 + 16 \), we can substitute \( x = -4 \) into the equation to find \( f(-4) \):

\[
f(-4) = -(-4)^2 + 16
\]
\[
f(-4) = -(16) + 16
\]
\[
f(-4) = 0
\]

### Final Answers:
a) \( f(x) = -x^2 + 16 \)  
b) \( f(-4) = 0 \)

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Do you need more details or have any questions?

Here are 5 related questions for further exploration:
1. How does changing the value of \( a \) affect the shape of the parabola?
2. What happens to the vertex if we change the value of \( h \) in the equation?
3. Can you explain how to find the roots of the parabola \( f(x) = -x^2 + 16 \)?
4. How would the graph look if the parabola opened upwards instead?
5. How do you find the axis of symmetry for a parabola?

**Tip:** The vertex form of a quadratic function makes it easier to graph and identify key points like the vertex and direction of the parabola.

The graph shows a function f(x) that is a parabola. a) Determine f(x) (i.e., write the equation). b) Determine f(-4).

Learn how to determine the equation of a parabola from its graph and evaluate the function at a specific point. This problem involves finding the vertex form of a quadratic function and solving for f(-4). Suitable for students in Grades 9-10.

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