https://file.aiquickdraw.com/mathbot/file/1730061521517rmlfopu0.jpg

To find the standard form of the quadratic function \( y = ax^2 + bx + c \) given the graph, we need to determine the values of \( a \), \( b \), and \( c \).

### Observations:
1. **The value of \( c \):** 
   - The \( y \)-intercept (where \( x = 0 \)) is given as 3, so \( c = 3 \).

2. **The value of \( a \):**
   - Since the parabola opens downwards, \( a \) is negative.
   - We are given \( a = -1 \) in the answer space, so we assume \( a = -1 \).

3. **Finding \( b \):**
   - To find \( b \), we can use other points on the graph along with \( a = -1 \) and \( c = 3 \).

From the graph, it appears the vertex is located at \( (-3, 4) \).

### Vertex Form Conversion:
The vertex form of a quadratic function is:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex. Substituting \( a = -1 \), \( h = -3 \), and \( k = 4 \):
\[
y = -1(x + 3)^2 + 4
\]

Expanding this to get the standard form:
\[
y = -1(x^2 + 6x + 9) + 4
\]
\[
y = -x^2 - 6x - 9 + 4
\]
\[
y = -x^2 - 6x - 5
\]

### Final Answer:
The standard form of the quadratic function is:
\[
y = -x^2 - 6x - 5
\]

Thus:
- \( a = -1 \)
- \( b = -6 \)
- \( c = 3 \)

Would you like further details on this solution? 

### Related Questions:
1. How do you determine the vertex of a parabola?
2. How can you tell if a parabola opens upwards or downwards?
3. What are the steps to convert from vertex form to standard form?
4. How does the value of \( a \) affect the width of a parabola?
5. How do you find the \( y \)-intercept from a graph?

### Tip:
To quickly find \( b \) in standard form, identify the vertex and substitute points into the equation.

What is the standard form of the quadratic function y = ax^2 + bx + c shown in the graph below?

Learn how to determine the standard form of a quadratic function y = ax^2 + bx + c using a graph. This problem covers key concepts like identifying the vertex, determining the y-intercept, and converting from vertex form to standard form. Suitable for Grades 8-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation