Question

choose the correct choice from each drop down menu to make the statements true.

Explanation:

Step1: Analyze \( f(x) \)

Find two points on \( f(x) \). From the graph, when \( x = -4 \), \( f(-4)=5 \); when \( x = -2 \), \( f(-2)=-5 \). The slope \( m_f=\frac{-5 - 5}{-2-(-4)}=\frac{-10}{2}=-5 \). The y - intercept: when \( x = 0 \), from the graph, \( f(0)=0 \)? Wait, no, let's re - check. Wait, the first graph: when \( x=-4 \), the point is \( (-4,5) \), when \( x = - 2 \), the point is \( (-2,-5) \)? Wait, no, the arrow is at the bottom, maybe another point. Wait, actually, let's take \( x=-4 \), \( y = 5 \) and \( x=-3 \), \( y = 0 \)? Wait, no, the x - axis is crossed at \( x=-3 \)? Wait, maybe better to take two clear points. For \( f(x) \): let's take \( x=-4 \), \( y = 5 \) and \( x=-2 \), \( y=-5 \). So slope \( m_f=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-5 - 5}{-2-(-4)}=\frac{-10}{2}=-5 \). The equation of \( f(x) \): using point - slope form \( y - y_1=m(x - x_1) \), with \( (x_1,y_1)=(-4,5) \), \( y - 5=-5(x + 4) \), \( y=-5x-20 + 5=-5x-15 \). Wait, when \( x = 0 \), \( y=-15 \)? No, that doesn't match the graph. Wait, maybe I misread the graph. Let's look again. The first graph: the line goes from \( (-4,5) \) down to \( (-2,-5) \)? Wait, no, the grid lines: each square is 1 unit. So when \( x=-4 \), the y - value is 5 (since it's at the 5th grid line above x - axis), when \( x=-3 \), y = 0 (crossing x - axis), when \( x=-2 \), y=-5 (5th grid line below x - axis). So two points: \( (-4,5) \) and \( (-3,0) \). Then slope \( m_f=\frac{0 - 5}{-3-(-4)}=\frac{-5}{1}=-5 \). Equation: \( y-5=-5(x + 4) \), \( y=-5x-20 + 5=-5x-15 \). Wait, when \( x = 0 \), \( y=-15 \), but the graph at \( x = 0 \) seems to be at \( y = 0 \)? No, maybe I made a mistake. Let's check the second graph \( g(x) \). For \( g(x) \): when \( x = 0 \), \( y = 9 \) (since it's at the 9th grid line? Wait, no, the second graph's y - axis: the top point is at \( (0,9) \)? Wait, no, the grid lines: each square is 1 unit. So when \( x = 0 \), \( y = 9 \)? No, the first grid line above x - axis is 1, then 2, ..., 8, 9? Wait, the second graph: when \( x = 0 \), the y - value is 9 (the arrow is at (0,9)), and when \( x = 2 \), \( y = 0 \) (crossing x - axis at (2,0)). So two points for \( g(x) \): \( (0,9) \) and \( (2,0) \). Slope \( m_g=\frac{0 - 9}{2-0}=\frac{-9}{2}=-4.5 \). Wait, but maybe the problem is about comparing slopes or y - intercepts. Wait, the question is to choose from dropdowns (not shown), but assuming it's about slope or y - intercept. Let's re - evaluate \( f(x) \): let's take \( x=-4 \), \( y = 5 \); \( x=-2 \), \( y=-5 \). Slope \( m_f=\frac{-5 - 5}{-2-(-4)}=-5 \). For \( g(x) \): \( x = 0 \), \( y = 9 \); \( x = 2 \), \( y = 0 \). Slope \( m_g=\frac{0 - 9}{2-0}=-4.5 \). So \( |m_f|=5 \), \( |m_g| = 4.5 \), so \( f(x) \) has a steeper slope. The y - intercept of \( f(x) \): when \( x = 0 \), from the equation \( y=-5x-15 \), \( y=-15 \)? No, that can't be. Wait, maybe I misread the graph. Let's look at the first graph again. The line crosses the x - axis at \( x=-3 \), and when \( x=-4 \), \( y = 5 \), \( x=-3 \), \( y = 0 \), \( x=-2 \), \( y=-5 \), \( x=-1 \), \( y=-10 \)? No, that's too steep. Wait, maybe each grid square is 1 unit, so the first graph: the line goes from \( (-4,5) \) to \( (-2,-5) \), so the change in y is - 10, change in x is 2, so slope - 5. The second graph: from \( (0,9) \) to \( (2,0) \), change in y is - 9, change in x is 2, slope - 4.5. So if the question is about which line has a greater slope (in absolute value), \( f(x) \) has a slope of - 5, \( g(x) \) has a slope of - 4.5, so \( | - 5|>| - 4.5| \), so \( f(x) \) is steeper. If the question is about y - intercept: \( f(x) \) at \( x = 0 \), from the equation \( y=-5x-15 \), \( y=-15 \), but \( g(x) \) at \( x = 0 \) is 9. So \( g(x) \) has a greater y - intercept.

Step2: Analyze \( g(x) \)

As above, for \( g(x) \), two points: \( (0,9) \) and \( (2,0) \). Slope \( m_g=\frac{0 - 9}{2-0}=-4.5 \). Equation: \( y-9=-4.5(x - 0) \), \( y=-4.5x + 9 \). So y - intercept is 9, slope is - 4.5.

(Note: Since the actual dropdown options are not provided, we assume the common comparison of slope and y - intercept. If the question is about which function has a greater slope (in absolute value), \( f(x) \) has slope - 5, \( g(x) \) has slope - 4.5, so \( f(x) \) is steeper. If about y - intercept, \( g(x) \) has y - intercept 9, \( f(x) \) (from correct graph reading, maybe I made a mistake earlier) let's re - check \( f(x) \): when \( x=-4 \), y = 5; \( x=-3 \), y = 0; \( x=-2 \), y=-5; \( x=-1 \), y=-10; \( x = 0 \), y=-15. For \( g(x) \): \( x = 0 \), y = 9; \( x = 1 \), y = 9-4.5 = 4.5; \( x = 2 \), y = 0. So if the dropdowns are about "The slope of \( f(x) \) is [greater/less] than the slope of \( g(x) \) (in absolute value)" or "The y - intercept of \( f(x) \) is [greater/less] than the y - intercept of \( g(x) \)". Since \( |-5|=5 \) and \( |-4.5| = 4.5 \), \( 5>4.5 \), so slope of \( f(x) \) (absolute value) is greater. Y - intercept of \( f(x) \) is - 15, y - intercept of \( g(x) \) is 9, so - 15<9, so y - intercept of \( f(x) \) is less than that of \( g(x) \).)

Answer:

(Assuming the question is about slope (absolute value) and y - intercept: The slope of \( f(x) \) (in absolute value) is greater than the slope of \( g(x) \), and the y - intercept of \( g(x) \) is greater than the y - intercept of \( f(x) \). But since the dropdown options are not given, we can only provide the analysis. If the options were like "The slope of \( f(x) \) is [greater/less] than the slope of \( g(x) \)" and "The y - intercept of \( f(x) \) is [greater/less] than the y - intercept of \( g(x) \)", the answers would be "greater" for slope (absolute) and "less" for y - intercept (since \( f(x) \) has y - intercept - 15, \( g(x) \) has 9).)