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, let’s analyze each claim one by one:
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To determine which student’s claim about the function is correct, let’s analyze each claim one by one:
1. Jeremiah: The y-intercept is at (15, 0).
To find the y-intercept, we need to set x = 0 and solve for y. Plugging in x = 0 into the function, we get f(0) = (0 + 3)(0 + 5) = 15. Therefore, Jeremiah’s claim is correct. The y-intercept is indeed at (0, 15), not (15, 0).
2. Lindsay: The x-intercepts are at (-3, 0) and (5, 0).
To find the x-intercepts, we need to set y = 0 and solve for x. Setting f(x) = 0, we have (x + 3)(x + 5) = 0. This equation is satisfied when either x + 3 = 0 or x + 5 = 0. So, the x-intercepts are indeed at x = -3 and x = -5, not (-3, 0) and (5, 0). So, Lindsay’s claim is incorrect.
3. Stephen: The vertex is at (-4, -1).
The vertex of a quadratic function can be found by evaluating the x-coordinate as -b/2a, and then substituting that value back into the function to find the corresponding y-coordinate. Comparing the given function f(x) = (x + 3)(x + 5) to the standard quadratic form f(x) = ax^2 + bx + c, we can determine that a = 1, b = 8, and c = 15. Therefore, the x-coordinate of the vertex is -b/2a = -8/(2*1) = -4. To find the y-coordinate, we plug x = -4 into the function: f(-4) = (-4 + 3)(-4 + 5) = -1. Thus, Stephen’s claim is correct. The vertex is indeed at (-4, -1).
4. Alexis: The midpoint between the x-intercepts is at (4, 0).
To find the midpoint between two points, we average the x-coordinates and the y-coordinates separately. The x-coordinate of the first x-intercept is -3, and the x-coordinate of the second x-intercept is -5. Averaging them, we get (-3 + (-5))/2 = -4. Similarly, the y-coordinate of both x-intercepts is 0, so averaging them gives (0 + 0)/2 = 0. Therefore, Alexis’ claim is incorrect. The midpoint between the x-intercepts is (-4, 0), not (4, 0).
In conclusion, based on the analysis of each claim, Jeremiah’s claim that the y-intercept is at (0, 15) is correct. The other three claims made by Lindsay, Stephen, and Alexis are incorrect.
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