Let's analyze each student's claim about the quadratic function [tex]\( f(x) = (x + 3)(x + 5) \)[/tex].
1. Jeremiah's Claim: The [tex]\( y \)[/tex]-intercept is at [tex]\( (15,0) \)[/tex]
- To find the [tex]\( y \)[/tex]-intercept, we evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex].
[tex]\[
f(0) = (0 + 3)(0 + 5) = 3 \times 5 = 15
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( (0, 15) \)[/tex]. Jeremiah's claim is incorrect as he interchanged the coordinates.
2. Lindsay's Claim: The [tex]\( x \)[/tex]-intercepts are at [tex]\( (-3,0) \)[/tex] and [tex]\( (5,0) \)[/tex]
- To find the [tex]\( x \)[/tex]-intercepts, we solve [tex]\( f(x) = 0 \)[/tex].
[tex]\[
(x + 3)(x + 5) = 0
\][/tex]
This gives us [tex]\( x = -3 \)[/tex] and [tex]\( x = -5 \)[/tex]. Therefore, the [tex]\( x \)[/tex]-intercepts are at [tex]\( (-3,0) \)[/tex] and [tex]\( (-5,0) \)[/tex]. Lindsay's claim is incorrect because she gave [tex]\( (5,0) \)[/tex] instead of [tex]\( (-5,0) \)[/tex].
3. Stephen's Claim: The vertex is at [tex]\( (-4, -1) \)[/tex]
- The vertex of a parabola given in standard form [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula [tex]\(\left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \)[/tex].
Expanding [tex]\( f(x) \)[/tex] we get:
[tex]\[
(x + 3)(x + 5) = x^2 + 8x + 15
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 8 \)[/tex]. The [tex]\( x \)[/tex]-coordinate of the vertex is:
[tex]\[
x = -\frac{b}{2a} = -\frac{8}{2 \cdot 1} = -4
\][/tex]
Substitute [tex]\( x = -4 \)[/tex] into [tex]\( f(x) \)[/tex] to find the [tex]\( y \)[/tex]-coordinate:
[tex]\[
f(-4) = (-4 + 3)(-4 + 5) = (-1)(1) = -1
\][/tex]
Thus, the vertex is indeed at [tex]\( (-4, -1) \)[/tex]. Stephen's claim is correct.
4. Alexis's Claim: The midpoint between the [tex]\( x \)[/tex]-intercepts is at [tex]\( (4,0) \)[/tex]
- The midpoint between the [tex]\( x \)[/tex]-intercepts [tex]\((-3,0)\)[/tex] and [tex]\((-5,0)\)[/tex] is:
[tex]\[
\left( \frac{-3 + (-5)}{2}, 0 \right) = \left( \frac{-8}{2}, 0 \right) = (-4, 0)
\][/tex]
Alexis's claim is incorrect because the correct midpoint is [tex]\( (-4,0) \)[/tex] not [tex]\( (4,0) \)[/tex].
After analyzing all the claims, the correct one is:
The claim by Stephen is correct.
