Question

function p: p(x) = 2x + 1; function g: x | g(x)
0 | -2
1 | -4
2 | -6
3 | -8; function r: (graph of r(x) as a line); which statements correctly compare the slopes and y-intercepts of the three functions?

Explanation:

Step1: Find slope of \( p(x) \)

The function \( p(x) = 2x + 1 \) is in slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. So, the slope of \( p(x) \) is \( m_p=2 \) and the y - intercept \( b_p = 1 \).

Step2: Find slope of \( g(x) \)

For a linear function given by a table, the slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using two points from the table of \( g(x) \), say \( (x_1,y_1)=(0, - 2) \) and \( (x_2,y_2)=(1,-4) \). Then \( m_g=\frac{-4-(-2)}{1 - 0}=\frac{-4 + 2}{1}=-2 \). The y - intercept of \( g(x) \) is the value of \( g(x) \) when \( x = 0 \), so \( b_g=-2 \).

Step3: Find slope of \( r(x) \)

The graph of \( r(x) \) is a line passing through the origin \( (0,0) \) (since it crosses the x - axis and y - axis at \( (0,0) \)) and another point, say, from the graph, if we take two points, for example, when \( x = 1 \), \( y=-1 \) (by looking at the grid). The slope \( m_r=\frac{-1-0}{1 - 0}=-1 \). The y - intercept of \( r(x) \) is \( 0 \) (since it passes through \( (0,0) \)).

Now, we can compare the slopes (\( m_p = 2\), \( m_g=-2\), \( m_r=-1\)) and y - intercepts (\( b_p = 1\), \( b_g=-2\), \( b_r = 0\)). For example, the slope of \( p(x) \) is positive, while the slopes of \( g(x) \) and \( r(x) \) are negative. The y - intercept of \( p(x) \) is positive, the y - intercept of \( g(x) \) is negative, and the y - intercept of \( r(x) \) is zero.

(Note: Since the question is about comparing slopes and y - intercepts, after calculating the slopes and y - intercepts of each function, we can make statements like "The slope of \( p(x) \) is greater than the slopes of \( g(x) \) and \( r(x) \)" or "The y - intercept of \( p(x) \) is positive, the y - intercept of \( g(x) \) is negative, and the y - intercept of \( r(x) \) is zero" etc. depending on the actual options (which are not fully visible, but the process of finding slopes and y - intercepts is as above).)

Answer:

(The final answer will depend on the specific comparison statements in the options. But the process to find slopes and y - intercepts is as follows: Slope of \( p(x)=2 \), y - intercept \( = 1 \); Slope of \( g(x)=-2 \), y - intercept \(=-2 \); Slope of \( r(x)=-1 \), y - intercept \( = 0 \))