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

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

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

Answer

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 ))