Question

a quadratic function, f(x), is described by the equation f(x) = x² + 4x - 10
some of the values of a second quadratic function, g(x), are shown in the table below.

x	-6	-4	-2	0	2	4
g(x)	0	-12	-16	-12	0	20

which statement is a true comparison of the properties of f(x) and g(x)?
a. the graph of g(x) has a higher minimum value and a higher y-intercept value than the graph of f(x).
b. the graph of g(x) has a higher minimum value but a lower y-intercept value than the graph of f(x).
c. the graph of g(x) has a lower minimum value and a lower y-intercept value than the graph of f(x).
d. the graph of g(x) has a lower minimum value but a higher y-intercept value than the graph of f(x).

Explanation:

Step1: Analyze \( f(x) \)

The function \( f(x) = x^2 + 4x - 10 \) is a quadratic function. We can rewrite it in vertex form to find the minimum value. Completing the square:
\[ \begin{align*} f(x)&=x^2 + 4x - 10\\ &=x^2 + 4x + 4 - 4 - 10\\ &=(x + 2)^2 - 14 \end{align*} \]
So the vertex of \( f(x) \) is at \( (-2, -14) \), and the minimum value (since the coefficient of \( x^2 \) is positive) is \( -14 \). The \( y \)-intercept of \( f(x) \) is found by setting \( x = 0 \): \( f(0) = 0^2 + 4(0) - 10 = -10 \).

Step2: Analyze \( g(x) \)

From the table of \( g(x) \), we can find the vertex (minimum point) by looking for symmetry. The values of \( g(x) \) are symmetric around \( x = -2 \) (since \( g(-6) = g(0) = 0 \), \( g(-4) = g(-12) \), etc.). The minimum value of \( g(x) \) occurs at \( x = -2 \), and \( g(-2) = -16 \)? Wait, no, looking at the table: when \( x = -6 \), \( g(x) = 0 \); \( x = -4 \), \( g(x) = -12 \); \( x = -2 \), \( g(x) = -16 \); \( x = 0 \), \( g(x) = -12 \); \( x = 2 \), \( g(x) = 0 \); \( x = 4 \), \( g(x) = 20 \). Wait, actually, the minimum value of \( g(x) \) is \( -16 \) at \( x = -2 \)? Wait, no, let's check again. Wait, the vertex of a quadratic is the minimum (since the parabola opens upwards, as the values increase on both sides of \( x = -2 \)). Wait, when \( x \) moves from \( -6 \) to \( -2 \), \( g(x) \) decreases from \( 0 \) to \( -16 \), then increases from \( -16 \) to \( 20 \) as \( x \) moves from \( -2 \) to \( 4 \). So the minimum value of \( g(x) \) is \( -16 \)? Wait, no, wait the table: \( x = -6 \), \( g(x) = 0 \); \( x = -4 \), \( g(x) = -12 \); \( x = -2 \), \( g(x) = -16 \); \( x = 0 \), \( g(x) = -12 \); \( x = 2 \), \( g(x) = 0 \); \( x = 4 \), \( g(x) = 20 \). Wait, no, the \( g(x) \) values at \( x = 2 \) is \( 0 \)? Wait the table says: \( x \): -6, -4, -2, 0, 2, 4; \( g(x) \): 0, -12, -16, -12, 0, 20. So the vertex is at \( x = -2 \), and \( g(-2) = -16 \). Wait, but the minimum value of \( f(x) \) is \( -14 \), and the minimum value of \( g(x) \) is \( -16 \)? Wait, no, that can't be. Wait, no, I must have made a mistake. Wait, \( f(x) \) has a minimum of \( -14 \), and \( g(x) \) has a minimum of \( -16 \)? But that would mean \( g(x) \) has a lower minimum. But wait, let's check the \( y \)-intercept of \( g(x) \). The \( y \)-intercept of \( g(x) \) is when \( x = 0 \), so \( g(0) = -12 \). Wait, \( f(0) = -10 \), so \( g(0) = -12 \), which is lower than \( f(0) = -10 \)? Wait, no, wait the table says \( g(0) = -12 \)? Wait the table: \( x = 0 \), \( g(x) = -12 \)? Wait the original table:

| \( x \) | -6 | -4 | -2 | 0 | 2 | 4 |
| --- | --- | --- | --- | --- | --- | --- |
| \( g(x) \) | 0 | -12 | -16 | -12 | 0 | 20 |

Ah, yes. So \( g(0) = -12 \). Now, the minimum value of \( f(x) \) is \( -14 \) (from vertex form \( (x + 2)^2 - 14 \)), and the minimum value of \( g(x) \) is \( -16 \) (at \( x = -2 \)). Wait, but that would mean \( g(x) \) has a lower minimum. But wait, let's re-express \( f(x) \) correctly. \( f(x) = x^2 + 4x - 10 \). The vertex form is \( (x + 2)^2 - 14 \), so the minimum value is \( -14 \). For \( g(x) \), the minimum value is \( -16 \) (at \( x = -2 \)). Now, the \( y \)-intercept of \( f(x) \) is \( f(0) = -10 \), and the \( y \)-intercept of \( g(x) \) is \( g(0) = -12 \). Wait, but that contradicts. Wait, no, maybe I made a mistake in the vertex of \( g(x) \). Wait, let's check the symmetry again. The points \( (-6, 0) \) and \( (0, -12) \)? No, wait, \( g(-6) = 0 \), \( g(0) = -12 \)? No, the table says \( g(0) = -12 \)? Wait the table is:

\( x \): -6, \( g(x) \): 0

\( x \): -4, \( g(x) \): -12

\( x \): -2, \( g(x) \): -16

\( x \): 0, \( g(x) \): -12

\( x \): 2, \( g(x) \): 0

\( x \): 4, \( g(x) \): 20

Ah, so the axis of symmetry is \( x = -2 \) (since \( g(-6) = g(2) = 0 \), \( g(-4) = g(0) = -12 \)). So the vertex is at \( x = -2 \), and \( g(-2) = -16 \). So the minimum value of \( g(x) \) is \( -16 \), and the minimum value of \( f(x) \) is \( -14 \). So \( g(x) \) has a lower minimum. Now, the \( y \)-intercept of \( f(x) \) is \( f(0) = -10 \), and the \( y \)-intercept of \( g(x) \) is \( g(0) = -12 \)? Wait, no, \( g(0) = -12 \), which is lower than \( f(0) = -10 \)? But that can't be. Wait, no, wait the table says \( g(0) = -12 \)? Wait, maybe I misread the table. Let me check again. The table:

\( x \): -6, \( g(x) \): 0

\( x \): -4, \( g(x) \): -12

\( x \): -2, \( g(x) \): -16

\( x \): 0, \( g(x) \): -12

\( x \): 2, \( g(x) \): 0

\( x \): 4, \( g(x) \): 20

Yes, that's correct. So \( g(0) = -12 \), \( f(0) = -10 \). Now, let's check the options:

A. The graph of \( g(x) \) has a higher minimum value and a higher \( y \)-intercept value than the graph of \( f(x) \). No, because \( g(x) \)'s minimum is \( -16 \) (lower than \( -14 \)) and \( y \)-intercept is \( -12 \) (lower than \( -10 \)).

B. The graph of \( g(x) \) has a higher minimum value but a lower \( y \)-intercept value than the graph of \( f(x) \). No, minimum of \( g(x) \) is lower.

C. The graph of \( g(x) \) has a lower minimum value and a lower \( y \)-intercept value than the graph of \( f(x) \). Let's check: minimum of \( g(x) \) is \( -16 \) (lower than \( -14 \)), \( y \)-intercept of \( g(x) \) is \( -12 \) (lower than \( -10 \))? Wait, no, \( -12 \) is lower than \( -10 \)? Wait, \( -12 < -10 \), yes. So minimum of \( g(x) \) is \( -16 < -14 \), \( y \)-intercept of \( g(x) \) is \( -12 < -10 \). So this seems correct.

D. The graph of \( g(x) \) has a lower minimum value but a higher \( y \)-intercept value than the graph of \( f(x) \). No, \( y \)-intercept of \( g(x) \) is lower.

Wait, but wait, maybe I made a mistake in the minimum of \( g(x) \). Wait, let's re-express \( f(x) \) again. \( f(x) = x^2 + 4x - 10 \). The vertex is at \( x = -b/(2a) = -4/(2*1) = -2 \), and \( f(-2) = (-2)^2 + 4*(-2) - 10 = 4 - 8 - 10 = -14 \), correct. For \( g(x) \), the minimum is at \( x = -2 \), \( g(-2) = -16 \), correct. \( y \)-intercept of \( f(x) \) is \( -10 \), \( y \)-intercept of \( g(x) \) is \( g(0) = -12 \), correct. So \( g(x) \) has a lower minimum (\( -16 < -14 \)) and a lower \( y \)-intercept (\( -12 < -10 \)). So option C is correct.

Answer:

C. The graph of \( g(x) \) has a lower minimum value and a lower \( y \)-intercept value than the graph of \( f(x) \).