Question

1. use the graph ( y = f(x) ) below.

graph of ( y = f(x) ) on a coordinate plane

what is the graph of ( g(x) = \frac{1}{2}f(x) - 6 )?

four option graphs labeled a, b, c, d

Explanation:

Step1: Analyze Vertical Compression

The function \( g(x)=\frac{1}{2}f(x)-6 \) involves a vertical compression by a factor of \( \frac{1}{2} \) (since we multiply \( f(x) \) by \( \frac{1}{2} \)) and a vertical shift down by 6 units. First, consider the vertical compression: each \( y \)-value of \( f(x) \) is multiplied by \( \frac{1}{2} \).

Step2: Analyze Vertical Shift

After the compression, we subtract 6 from each \( y \)-value (vertical shift down 6 units). Let's check key points of \( f(x) \). For example, if \( f(x) \) has a point \( (x,y) \), then \( g(x) \) will have \( (x, \frac{1}{2}y - 6) \). Looking at the original graph \( y = f(x) \), let's take the y - intercept: from the original graph, the y - intercept of \( f(x) \) seems to be around \( y = 8 \) (estimating from the grid). Then for \( g(x) \), the y - intercept would be \( \frac{1}{2}(8)-6=4 - 6=-2 \)? Wait, no, maybe better to check the slope. The original line (the linear part) of \( f(x) \): let's find two points. From the original graph, the linear part goes from \( (0,8) \) to \( (8,0) \), so slope \( m=\frac{0 - 8}{8 - 0}=-1 \). So equation of linear part: \( y=-x + 8 \). Then \( g(x)=\frac{1}{2}(-x + 8)-6=-\frac{1}{2}x+4 - 6=-\frac{1}{2}x - 2 \). Now check the options. Let's look at the slope and y - intercept. The slope of \( g(x) \) is \( -\frac{1}{2} \), y - intercept \( - 2 \). Wait, maybe my initial y - intercept was wrong. Wait, maybe the original linear part: let's re - examine the original graph. The original graph's linear segment: when \( x = 0 \), \( y = 8 \)? Wait, no, looking at the first graph, the vertical axis: the grid lines, maybe each grid is 2 units? Wait, maybe the original graph's linear part: from \( (0,8) \) to \( (8,0) \) (if each grid is 2 units? No, maybe each grid is 1 unit. Wait, the x - axis has marks at - 16, - 12, - 8, - 4, 0, 4, 8, 12, 16. The y - axis has marks at - 12, - 8, - 4, 0, 4, 8, 12, 16, 20, 24. So the original linear part: let's take two points. Let's say at \( x = 0 \), \( y = 8 \); at \( x = 8 \), \( y = 0 \). So \( f(x) \) (linear part) is \( y=-x + 8 \). Then \( g(x)=\frac{1}{2}f(x)-6=\frac{1}{2}(-x + 8)-6=-\frac{1}{2}x + 4-6=-\frac{1}{2}x - 2 \). Now check the options. Let's look at option b: the linear part. Let's check the slope and y - intercept. The equation \( y =-\frac{1}{2}x - 2 \) has slope \( -\frac{1}{2} \) and y - intercept \( - 2 \). Looking at option b, the linear part seems to have slope \( -\frac{1}{2} \) and y - intercept around - 2? Wait, maybe I made a mistake. Wait, maybe the original y - intercept is different. Wait, maybe the original graph's linear part is from \( (0, 8) \) to \( (8,0) \), but when we do \( g(x)=\frac{1}{2}f(x)-6 \), the vertical compression: each y - value is halved, then shifted down 6. So for the linear part, the slope of \( f(x) \) is - 1, so slope of \( g(x) \) is \( \frac{1}{2}\times(-1)=-\frac{1}{2} \) (since vertical compression affects the slope's magnitude? Wait, no: if \( y = f(x) \) has slope \( m \), then \( y=\frac{1}{2}f(x) \) has slope \( \frac{1}{2}m \), because \( \frac{1}{2}f(x) \) is a vertical scaling. So original slope \( m=-1 \), so new slope \( m'=\frac{1}{2}\times(-1)=-\frac{1}{2} \). The y - intercept of \( f(x) \): from the original graph, when \( x = 0 \), \( y = 8 \) (let's assume). Then \( \frac{1}{2}\times8-6 = 4 - 6=-2 \). So the equation of \( g(x) \) for the linear part is \( y =-\frac{1}{2}x-2 \). Now check the options. Let's look at option b: the graph in option b has a linear part with slope \( -\frac{1}{2} \) and y - intercept around - 2? Wait, maybe the correct option is b? Wait, no, let's check the other parts (the non - linear parts, the zig - zag). The non - linear parts (the triangular waves) will also be vertically compressed by \( \frac{1}{2} \) and shifted down 6. So the height of the triangles will be halved, and shifted down. Looking at the options, option b seems to have the linear part with slope \( -\frac{1}{2} \) and the correct vertical shift and compression.

Answer:

b (assuming the option b is the one with the linear part \( y =-\frac{1}{2}x - 2 \) and the compressed triangular waves shifted down)