https://file.aiquickdraw.com/mathbot/file/17253828871291u9ito8x.jpg

To solve this problem, we need to identify which of the given equations matches the graph of the function \( f(x) \) shown.

### Step 1: Identify key features of the graph
- **Vertex**: The graph has its vertex at the point \( (1, -9) \).
- **Direction**: The graph opens upwards, which indicates that the leading coefficient of \( x^2 \) is positive.
- **X-Intercepts**: The graph crosses the x-axis at \( x = -2 \) and \( x = 4 \).

### Step 2: Analyze the given options
1. **Option A**: \( f(x) = x^2 - 2x - 8 \)
   - This is in standard form \( ax^2 + bx + c \).
   - To find the vertex, use the vertex formula \( x = -\frac{b}{2a} = -\frac{-2}{2 \times 1} = 1 \).
   - Plugging \( x = 1 \) into the equation: \( f(1) = 1^2 - 2(1) - 8 = 1 - 2 - 8 = -9 \). 
   - The vertex is \( (1, -9) \), which matches the graph.
   - To check the x-intercepts, set \( f(x) = 0 \):
     \[
     x^2 - 2x - 8 = 0
     \]
     Factor the equation:
     \[
     (x - 4)(x + 2) = 0
     \]
     So the x-intercepts are \( x = 4 \) and \( x = -2 \), which match the graph.

2. **Option B**: \( f(x) = -x^2 + 2x - 8 \)
   - This opens downwards due to the negative sign in front of \( x^2 \), which does not match the graph.

3. **Option C**: \( f(x) = (x - 2)(x + 4) \)
   - This factored form has roots at \( x = 2 \) and \( x = -4 \), which do not match the graph's intercepts.

4. **Option D**: \( f(x) = -(x - 1)^2 - 9 \)
   - This is in vertex form with the vertex at \( (1, -9) \), but it opens downwards (due to the negative sign in front), so it does not match the graph.

### Conclusion
The correct equation that matches the graph is **Option A: \( f(x) = x^2 - 2x - 8 \)**.

Would you like more details on any specific part of the solution? Here are some related questions you might find helpful:

1. How do you convert a quadratic equation from standard form to vertex form?
2. What are the properties of parabolas based on the sign of the leading coefficient?
3. How do you find the x-intercepts of a quadratic function?
4. What is the significance of the vertex in the graph of a quadratic function?
5. How can you determine if a parabola opens upwards or downwards?

**Tip**: To quickly determine the vertex of a parabola in standard form, use the formula \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex.

The graph of y = f(x) is shown in the xy-plane below. Which of the following equations could define f(x)? Options: A) f(x) = x^2 - 2x - 8 B) f(x) = -x^2 + 2x - 8 C) f(x) = (x - 2)(x + 4) D) f(x) = -(x - 1)^2 - 9

This problem involves analyzing a quadratic graph with a vertex at (1, -9) and determining the matching equation among the options. It covers key concepts in quadratic functions, including identifying the vertex and x-intercepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation