Analyzing Claims About a Math Function: Determining Correct 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.

Stephen

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

Jeremiah’s claim: The y-intercept is at (15, 0).
To find the y-intercept, we need to substitute x = 0 into the function and solve for y:
f(0) = (0 + 3)(0 + 5)
f(0) = 3 × 5
f(0) = 15
So, the y-intercept is at (0, 15), not (15, 0). Jeremiah’s claim is incorrect.

Lindsay’s claim: The x-intercepts are at (-3, 0) and (5, 0).
To find the x-intercepts, we set y = 0 in the function and solve for x:
(x + 3)(x + 5) = 0
Either x + 3 = 0 or x + 5 = 0
If x + 3 = 0, x = -3 (giving us the x-intercept at (-3, 0))
If x + 5 = 0, x = -5 (not mentioned by Lindsay)
So, Lindsay’s claim is partially correct; there is an x-intercept at (-3, 0), but not at (5, 0).

Stephen’s claim: The vertex is at (-4, -1).
To find the vertex of the function f(x) = (x + 3)(x + 5), we first need to rewrite it in standard form (vertex form):
f(x) = x^2 + (3 + 5)x + (3 × 5)
f(x) = x^2 + 8x + 15
The x-coordinate of the vertex in standard form is given by x = -b / (2a)
In this case, a = 1 and b = 8, so x = -8 / (2 × 1) = -4.
To find y-coordinate, substitute x = -4 into the function:
f(-4) = (-4)^2 + 8(-4) + 15
f(-4) = 16 – 32 + 15
f(-4) = -1
So, Stephen’s claim is correct; the vertex is at (-4, -1).

Alexis’s claim: The midpoint between the x-intercepts is at (4, 0).
To find the midpoint between two points, we need to average their x and y coordinates.
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
In this case, the x-intercepts are at (-3, 0) and (5, 0).
Midpoint = ((-3 + 5) / 2, (0 + 0) / 2)
Midpoint = (2 / 2, 0)
Midpoint = (1, 0)
So, Alexis’s claim is incorrect; the midpoint between the x-intercepts is at (1, 0), not (4, 0).

Based on the analysis, Stephen’s claim that the vertex is at (-4, -1) is the correct one.

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