Question

which statement is true regarding the graphed functions?
○ $f(0) = 2$ and $g(-2) = 0$
○ $f(0) = 4$ and $g(-2) = 4$
○ $f(2) = 0$ and $g(-2) = 0$
○ $f(-2) = 0$ and $g(-2) = 0$

Explanation:

Step1: Analyze \( f(0) \)

To find \( f(0) \), look at the graph of \( f(x) \) (blue parabola) at \( x = 0 \). The point where \( x = 0 \) on \( f(x) \) has a \( y \)-value of 4. So \( f(0)=4 \).

Step2: Analyze \( g(-2) \)

To find \( g(-2) \), look at the graph of \( g(x) \) (red parabola) at \( x=-2 \). The vertex of \( g(x) \) is at \( x = -2 \), and the \( y \)-value there is 0? Wait, no, wait. Wait, also check \( g(-2) \) and \( f(0) \) against options. Wait, also check the intersection. Wait, the red graph \( g(x) \) at \( x=-2 \): wait, no, let's recheck. Wait, the blue graph \( f(x) \) at \( x=0 \) is (0,4). The red graph \( g(x) \): let's check \( x=-2 \), the red graph's vertex is at \( x=-2 \), \( y=0 \)? Wait no, wait the options: let's check each option.

Wait, first, \( f(0) \): the blue parabola (f(x)) at x=0 is (0,4), so \( f(0)=4 \). Then check \( g(-2) \): the red parabola (g(x)) at x=-2: wait, no, wait the red parabola's vertex is at x=-2, y=0? Wait no, looking at the graph, the red parabola has a vertex at x=-2, y=0? Wait no, the grid: the red parabola at x=-2, the y-coordinate: looking at the graph, the red parabola touches the x-axis at x=-2, so g(-2)=0? Wait no, wait the options: let's check the second option: \( f(0)=4 \) and \( g(-2)=4 \)? Wait no, wait maybe I made a mistake. Wait, the two parabolas intersect at (0,4). So f(0)=4. Now check g(-2): the red parabola (g(x)) at x=-2: the vertex is at x=-2, y=0? Wait no, the red parabola at x=-2: looking at the graph, the red parabola's lowest point (vertex) is at x=-2, y=0? Wait, no, the grid lines: from the graph, the red parabola at x=-2, the y-coordinate is 0? Wait, but the second option says \( g(-2)=4 \). Wait, no, maybe I messed up. Wait, let's check the options again.

Wait the options:

Option 2: \( f(0)=4 \) and \( g(-2)=4 \). Wait, no, wait the red graph at x=-2: wait, no, the red graph at x=-2: wait, the red graph is a parabola opening upwards with vertex at x=-2, y=0? Wait, no, the y-axis: the red graph at x=-2, the y-value: looking at the graph, the red graph at x=-2 is at y=0? But the blue graph at x=0 is y=4. Wait, but the second option says \( f(0)=4 \) and \( g(-2)=4 \). Wait, maybe I made a mistake. Wait, let's check the red graph at x=-2: no, wait the red graph's vertex is at x=-2, y=0, but the blue graph at x=0 is y=4. Wait, but the second option is \( f(0)=4 \) and \( g(-2)=4 \). Wait, maybe I misread the graph. Wait, the red graph: at x=-2, is the y-value 4? No, that can't be. Wait, maybe the red graph at x=-2: no, wait the two parabolas: f(x) is blue, g(x) is red. The blue parabola has vertex at (2,0), red at (-2,0). So f(0)=4 (since it passes through (0,4)), g(-2)=0 (since it's at vertex ( -2,0)). But that's not matching. Wait, no, wait the options: let's check each option:

1. \( f(0)=2 \) and \( g(-2)=0 \): f(0) is 4, so wrong.

2. \( f(0)=4 \) and \( g(-2)=4 \): Wait, maybe the red graph at x=-2: no, wait maybe the red graph at x=-2 is 4? No, that doesn't make sense. Wait, maybe I messed up the graphs. Wait, the blue graph (f(x)): vertex at (2,0), opens upwards. The red graph (g(x)): vertex at (-2,0), opens upwards. They intersect at (0,4). So f(0)=4. Now, g(-2): the vertex of g(x) is at (-2,0), so g(-2)=0. But that's not in option 2. Wait, option 2 is \( f(0)=4 \) and \( g(-2)=4 \). Wait, maybe the red graph at x=-2 is 4? No, that's not possible. Wait, maybe the question has a typo, or I misread. Wait, let's check the third option: \( f(2)=0 \) and \( g(-2)=0 \). f(2) is the vertex of f(x), which is (2,0), so f(2)=0. g(-2) is the vertex of g(x), which is (-2,0), so g(-2)=0. Wait, that's option 3: \( f(2)=0 \) and \( g(-2)=0 \). Wait, but earlier I thought f(0)=4, but maybe I confused x=0 and x=2. Wait, f(2): the blue parabola (f(x)) at x=2 is (2,0), so f(2)=0. g(-2): the red parabola (g(x)) at x=-2 is ( -2,0), so g(-2)=0. So option 3: \( f(2)=0 \) and \( g(-2)=0 \). Wait, but earlier I thought f(0)=4, but that's for x=0. So let's re-express:

Wait, f(x) is the blue parabola, vertex at (2,0), so f(2)=0. g(x) is the red parabola, vertex at (-2,0), so g(-2)=0. So option 3: \( f(2)=0 \) and \( g(-2)=0 \) is correct? Wait, but let's check the options again:

Options:

1. \( f(0)=2 \) and \( g(-2)=0 \): f(0) is 4, so wrong.

2. \( f(0)=4 \) and \( g(-2)=4 \): g(-2) is 0, so wrong.

3. \( f(2)=0 \) and \( g(-2)=0 \): f(2) is 0 (vertex of blue), g(-2) is 0 (vertex of red), so this is correct.

4. \( f(-2)=0 \) and \( g(-2)=0 \): f(-2) is not 0 (blue parabola at x=-2 is above x-axis), so wrong.

Ah, I see, I made a mistake earlier by checking x=0 instead of x=2 for f(x). So f(2) is 0 (vertex of blue parabola at x=2, y=0), and g(-2) is 0 (vertex of red parabola at x=-2, y=0). So option 3 is correct.

Wait, but let's confirm:

- f(x) (blue) has vertex at (2,0), so f(2)=0.

- g(x) (red) has vertex at (-2,0), so g(-2)=0.

So the third option: \( f(2)=0 \) and \( g(-2)=0 \) is correct.

Answer:

\( \boldsymbol{f(2) = 0} \) and \( \boldsymbol{g(-2) = 0} \) (the third option: \( f(2) = 0 \) and \( g(-2) = 0 \))