https://file.aiquickdraw.com/mathbot/file/1728259544546evq71y1c.jpg

To find the quadratic function that represents the given graph, let’s analyze the provided information.

### Step-by-Step Solution:

1. **Vertex Form of a Quadratic Function**:
   The vertex form of a quadratic function is given by:

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

   where:
   - \( (h, k) \) is the vertex of the parabola.
   - \( a \) determines the shape and direction (upward or downward) of the parabola.

2. **Identify the Vertex**:
   From the graph, the vertex is clearly labeled as \( (-3, -4) \). So, \( h = -3 \) and \( k = -4 \).

3. **Substitute the Vertex into the Equation**:
   The equation in vertex form becomes:

   \[
   f(x) = a(x + 3)^2 - 4
   \]

4. **Find the Value of \( a \)**:
   To find \( a \), use another point on the graph. From the graph, a clear point is \( (0, 5) \).

   Plugging in \( x = 0 \) and \( f(0) = 5 \):

   \[
   5 = a(0 + 3)^2 - 4
   \]
   Simplifying:

   \[
   5 = a(9) - 4
   \]
   \[
   5 + 4 = 9a
   \]
   \[
   9 = 9a
   \]
   \[
   a = 1
   \]

5. **Final Equation**:
   The equation of the quadratic function is:

   \[
   f(x) = (x + 3)^2 - 4
   \]

This is the quadratic function that represents the given graph.

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

1. How would the graph change if \( a \) were negative?
2. What is the significance of the vertex in a quadratic function?
3. How do we convert from the vertex form to the standard form of a quadratic equation?
4. What other points on the graph can we use to verify \( a = 1 \)?
5. How does changing \( h \) or \( k \) affect the position of the parabola?

**Tip**: Always use a clear point on the graph, preferably an intercept, to solve for \( a \) accurately.

Determine the quadratic function whose graph is given with vertex (-3, -4) and point (0, 5).

Learn how to find the quadratic function given a graph with a vertex at (-3, -4) and point (0, 5). This solution demonstrates how to use the vertex form of a quadratic function to derive the equation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation