Analyzing Claims about a Quadratic Function | Determining Correctness of Various Statements

Mr. Walker gave his class the function f(x) = (x + 3)(x + 5). Four students made a claim about the function. Each student’s claim is below.Jeremiah: The y-intercept is at (15, 0).Lindsay: The x-intercepts are at (-3, 0) and (5, 0).Stephen: The vertex is at (-4, -1).Alexis: The midpoint between the x-intercepts is at (4, 0).Which student’s claim about the function is correct?The claim by is correct.

To determine which student’s claim about the function is correct, we can analyze each claim one by one

To determine which student’s claim about the function is correct, we can analyze each claim one by one.

1. Jeremiah: The y-intercept is at (15, 0).
To find the y-intercept, we substitute x = 0 into the function f(x). Thus, f(0) = (0 + 3)(0 + 5) = 3 * 5 = 15. Therefore, Jeremiah’s claim is correct, and the y-intercept is at (0, 15).

2. Lindsay: The x-intercepts are at (-3, 0) and (5, 0).
To find the x-intercepts, we set f(x) = 0 and solve for x.
(x + 3)(x + 5) = 0
We have two possibilities:
i) (x + 3) = 0, which gives x = -3
ii) (x + 5) = 0, which gives x = -5

Therefore, Lindsay’s claim is incorrect. The correct x-intercepts are (-3, 0) and (-5, 0).

3. Stephen: The vertex is at (-4, -1).
To find the vertex of a quadratic function in the form f(x) = ax^2 + bx + c, the x-coordinate of the vertex is given by -b/2a. In this case, a = 1, b = 8, and c = 15. Therefore, the x-coordinate of the vertex is -8/(2*1) = -4. However, the y-coordinate of the vertex is not -1 but is given by f(-4). We can compute this as f(-4) = (-4 + 3)(-4 + 5) = -1 * 1 = -1. So Stephen’s claim is correct, and the vertex is at (-4, -1).

4. Alexis: The midpoint between the x-intercepts is at (4, 0).
To find the midpoint between two points, we take the average of their x-coordinates and the average of their y-coordinates. In this case, the x-coordinates of the x-intercepts are -3 and 5. The average of these is (-3 + 5)/2 = 2/2 = 1. Therefore, Alexis’ claim is incorrect, and the midpoint between the x-intercepts is not (4, 0) but (1, 0).

In analyzing the claims made by each student, we found that Jeremiah’s claim about the y-intercept is correct, Lindsay’s claim about the x-intercepts is incorrect, Stephen’s claim about the vertex is correct, and Alexis’ claim about the midpoint between the x-intercepts is incorrect. Therefore, the student with the correct claim is Stephen.

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