https://file.aiquickdraw.com/mathbot/file/17248354690239gqldin3.jpg

The graph of the function y=−(x+a)2+b
is shown.

a=
Answer
, b=
Answer

The graph shows the function \( y = -(x + a)^2 + b \), which is a downward-opening parabola. To find the values of \( a \) and \( b \), we can use the vertex and y-intercept information given in the graph.

### Step 1: Identify the Vertex
The vertex of the parabola is given at the point \( (3, 5) \). The vertex form of a parabola is \( y = -(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. In our equation \( y = -(x + a)^2 + b \), we see that:

\[
h = -a \quad \text{and} \quad k = b
\]

Given that the vertex \( (h, k) = (3, 5) \):

\[
- a = 3 \quad \text{so} \quad a = -3
\]
\[
b = 5
\]

### Step 2: Verification Using the Y-intercept
The graph also shows that the parabola intersects the y-axis at \( (0, -4) \). Substituting \( x = 0 \) into the equation:

\[
y = -(0 + 3)^2 + 5
\]
\[
y = -(9) + 5 = -9 + 5 = -4
\]

This matches the y-intercept provided, confirming our values.

### Final Answer:
- \( a = -3 \)
- \( b = 5 \)

Would you like more details or have any other questions?

Here are some related questions:
1. How would the graph change if \( a \) were positive instead of negative?
2. What is the significance of the y-intercept in the graph of a parabola?
3. Can you explain how the direction of the parabola is determined from the equation?
4. How would you find the roots of this parabola?
5. What effect would altering the value of \( b \) have on the graph?

**Tip:** When working with the vertex form of a parabola, always identify the vertex first, as it directly gives you the values of \( h \) and \( k \).

Explore how to solve the quadratic function y = -(x + a)^2 + b with detailed step-by-step analysis. This problem involves finding the vertex form of a parabola, determining values of a and b from graph points, and verifying through the y-intercept. Suitable for high school students.

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