## 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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