https://file.aiquickdraw.com/mathbot/file/1726443458603p3h74ms2.jpg

The graph shows a parabola that opens downward. Let's break down the components of the function step-by-step:

### 1. **Identifying the Parent Function:**
   The shape of the graph indicates that it is a transformation of the basic quadratic parent function:
   \[
   f(x) = x^2
   \]
   However, the parabola is reflected downward and vertically compressed. This suggests the function is of the form:
   \[
   f(x) = -a(x - h)^2 + k
   \]
   where:
   - \( a \) affects the width and direction of the parabola.
   - \( (h, k) \) is the vertex of the parabola.

### 2. **Finding the Vertex:**
   From the graph, the vertex is at the point \( (0, 0) \). Thus, \( h = 0 \) and \( k = 0 \).

### 3. **Determining the Value of \( a \):**
   To find \( a \), we can use another point on the graph. From observation, the point \( (2, -2) \) lies on the parabola. Using this in the equation \( f(x) = -a(x)^2 \):
   \[
   -2 = -a(2)^2
   \]
   \[
   -2 = -4a
   \]
   Solving for \( a \):
   \[
   a = \frac{1}{2}
   \]
   Hence, the equation of the function is:
   \[
   f(x) = -\frac{1}{2}x^2
   \]

### 4. **Domain and Range:**
   - **Domain**: Since it's a quadratic function, the domain is all real numbers:
     \[
     \text{Domain} = (-\infty, \infty)
     \]
   - **Range**: The parabola opens downward and the vertex is at \( y = 0 \). Hence, the range is:
     \[
     \text{Range} = (-\infty, 0]
     \]

### Final Answer:
   The equation is:
   \[
   f(x) = -\frac{1}{2}x^2
   \]
   The domain is \( (-\infty, \infty) \), and the range is \( (-\infty, 0] \).

Would you like more details or have any questions?

Here are 5 relative questions:
1. How does the value of \( a \) affect the shape and direction of a parabola?
2. What is the general form of a quadratic equation in vertex form?
3. How would you find the equation of a parabola if the vertex were not at the origin?
4. What are the steps to determine the range of a quadratic function?
5. How can you tell if a parabola opens upwards or downwards?

**Tip**: Always verify the vertex and an additional point on the graph to find the correct equation of a parabola.

Identify the parent function f(x) and write an equation for the domain and range. The graph shows a parabola that opens downward.

This problem involves identifying a quadratic parent function, determining its equation, and finding the domain and range. The equation is in vertex form f(x) = -1/2x^2, with the domain (-∞, ∞) and the range (-∞, 0]. Suitable for high school students studying algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation