Question

parabola has a maximum value of 4 at x = -1, a y-intercept of 3, and an x-intercept of 1.

Which graph matches the description?

Answer 1

To determine which graph matches the description of the parabola, we can gather the key information provided:

1. **Maximum Value**: The parabola has a maximum value of 4 at \( x = -1 \). This means the vertex of the parabola is at the point \( (-1, 4) \).

2. **Y-Intercept**: The y-intercept of the parabola is 3. This means when \( x = 0 \), the value of \( y \) is 3.

3. **X-Intercept**: The x-intercept is 1, meaning when \( y = 0 \), \( x = 1 \).

Based on this information, we can write the general form of a parabola:

\[
y = a(x + 1)^2 + 4
\]

Here, the parabola opens downwards because it has a maximum value (the coefficient \( a \) will be negative).

Next, let's find the value of \( a \) using the y-intercept. If \( x = 0 \), we have:

\[
3 = a(0 + 1)^2 + 4
\]
\[
3 = a(1) + 4
\]
\[
3 = a + 4
\]
\[
a = 3 - 4 = -1
\]

Now we can write the equation of the parabola:

\[
y = -1(x + 1)^2 + 4
\]

Next, we find the x-intercept by setting \( y = 0 \):

\[
0 = -1(x + 1)^2 + 4
\]
\[
-1(x + 1)^2 = -4
\]
\[
(x + 1)^2 = 4
\]
\[
x + 1 = \pm 2
\]
\[
x = 1 \quad \text{or} \quad x = -3
\]

Thus, the parabola has x-intercepts at 1 and -3, confirming one of the x-intercepts is indeed at 1 as described.

Now we can summarize:
- Vertex is (-1, 4).
- Y-intercept is (0, 3).
- X-intercepts are (1, 0) and (-3, 0).

To find which graph matches this description, look for:
1. A downward-opening parabola with a vertex at (-1, 4).
2. A point on the y-axis at (0, 3).
3. A point on the x-axis at (1, 0).

This will help you identify the correct graph from the options provided.