Question

Sketch the quadratic function f(x) = x2+ 4x + 3. Which key feature of the graph is not correct?
Responses
A minimum (-2, -1)minimum (-2, -1)
B y-intercept (0, 2)y-intercept (0, 2)
C x-intercept (-3, 0)x-intercept (-3, 0)
D x-intercept (-1, 0)x-intercept (-1, 0)

Answer 1

To analyze the quadratic function \( f(x) = x^2 + 4x + 3 \), we can start by determining its features, such as its vertex, y-intercept, and x-intercepts.

1. **Finding the y-intercept**:
The y-intercept occurs when \( x = 0 \).
\[
f(0) = 0^2 + 4(0) + 3 = 3
\]
So, the y-intercept is \( (0, 3) \).

2. **Finding the x-intercepts**:
To find the x-intercepts, we set \( f(x) = 0 \):
\[
x^2 + 4x + 3 = 0
\]
We can factor this equation:
\[
(x + 1)(x + 3) = 0
\]
This gives us the x-intercepts:
\[
x + 1 = 0 \Rightarrow x = -1 \quad \Rightarrow (-1, 0)
\]
\[
x + 3 = 0 \Rightarrow x = -3 \quad \Rightarrow (-3, 0)
\]

3. **Finding the vertex**:
The vertex of a quadratic in standard form \( ax^2 + bx + c \) can be found at \( x = -\frac{b}{2a} \):
Here, \( a = 1 \) and \( b = 4 \):
\[
x = -\frac{4}{2 \cdot 1} = -2
\]
Now, we find \( f(-2) \):
\[
f(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1
\]
So, the vertex is \( (-2, -1) \), which is a minimum since the parabola opens upwards.

Now, let's summarize the features we found:
- Vertex: \( (-2, -1) \)
- Y-intercept: \( (0, 3) \)
- X-intercepts: \( (-1, 0) \) and \( (-3, 0) \)

Now, we can review the options provided:

A. Minimum \( (-2, -1) \) - Correct.
B. Y-intercept \( (0, 2) \) - Incorrect, should be \( (0, 3) \).
C. X-intercept \( (-3, 0) \) - Correct.
D. X-intercept \( (-1, 0) \) - Correct.

The key feature of the graph that is not correct is:
**B. y-intercept \( (0, 2) \)**. It should be \( (0, 3) \).