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