Question

which statements correctly compare the slopes and y-intercepts of the three functions?
the slope of function g is 2 times the slope of function r.
the slope of function r is 2 times the slope of function p.
the y-intercept of function p is less than the y-intercept of function r.
the y-intercept of function g is greater than the y-intercept of function p.
the y-intercept of r is 2 more than the y-intercept of g.

Explanation:

To solve this, we need to recall the general form of a linear function \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. However, since the actual functions \( p \), \( g \), and \( r \) are not given (maybe they are in a table or graph not shown here), we assume that we have the equations or data for these functions. Let's assume the following (common case for such problems):

Step 1: Analyze the slope statements

Suppose:
- Let the slope of function \( p \) be \( m_p \), slope of \( g \) be \( m_g \), slope of \( r \) be \( m_r \).
- If the statement "The slope of function \( g \) is 2 times the slope of function \( r \)" is \( m_g = 2m_r \), we need to check with the actual slopes.
- For "The slope of function \( r \) is 2 times the slope of function \( p \)", it would be \( m_r=2m_p \)

Step 2: Analyze the y - intercept statements

Let the y - intercept of \( p \) be \( b_p \), \( g \) be \( b_g \), \( r \) be \( b_r \)
- "The y - intercept of function \( p \) is less than the y - intercept of function \( r \)" means \( b_p < b_r \)
- "The y - intercept of function \( g \) is greater than the y - intercept of function \( p \)" means \( b_g>b_p \)
- "the y - intercept of \( r \) is 2 more than the y - intercept of \( g \)" means \( b_r=b_g + 2\)

Since the problem is about comparing slopes and y - intercepts of linear functions, we need to use the slope - intercept form \( y=mx + b \). But as the original problem's function definitions (like equations, tables, or graphs) are not fully provided, we can only give a general approach. However, if we assume a common problem setup (for example, if function \( p \) has slope \( m_p = 1\), \( r \) has \( m_r = 2\), \( g \) has \( m_g=4\); and y - intercepts \( b_p = 1\), \( b_r=3\), \( b_g = 1\)):

- For the slope of \( g \) (4) and \( r \) (2): \( 4=2\times2\), so "The slope of function \( g \) is 2 times the slope of function \( r \)" is correct.
- For the slope of \( r \) (2) and \( p \) (1): \( 2 = 2\times1\), so "The slope of function \( r \) is 2 times the slope of function \( p \)" is correct.
- For y - intercepts: If \( b_p = 1\), \( b_r = 3\), then \( 1<3 \), so "The y - intercept of function \( p \) is less than the y - intercept of function \( r \)" is correct.
- If \( b_g = 1\), \( b_p = 1\), then "The y - intercept of function \( g \) is greater than the y - intercept of function \( p \)" is false.
- If \( b_r=3\), \( b_g = 1\), then \( 3=1 + 2\), so "the y - intercept of \( r \) is 2 more than the y - intercept of \( g \)" is correct.

But since the problem is incomplete in terms of function definitions, we can only provide the method. However, in a typical problem of this type (found in textbooks), the correct statements are usually determined by calculating the slopes and y - intercepts from the given functions (equations, tables, or graphs).

Assuming that we have the correct analysis from the given (but not shown) functions:

If we consider a standard problem where:

- Function \( p\): \( y=x + 1\) (slope \( m_p = 1\), y - intercept \( b_p=1\))
- Function \( r\): \( y = 2x+3\) (slope \( m_r = 2\), y - intercept \( b_r = 3\))
- Function \( g\): \( y=4x + 1\) (slope \( m_g=4\), y - intercept \( b_g = 1\))

Then:

- The slope of \( g\) (4) is 2 times the slope of \( r\) (2) (since \( 4=2\times2\)): Correct.
- The slope of \( r\) (2) is 2 times the slope of \( p\) (1) (since \( 2 = 2\times1\)): Correct.
- The y - intercept of \( p\) (1) is less than the y - intercept of \( r\) (3): Correct.
- The y - intercept of \( g\) (1) is not greater than the y - intercept of \( p\) (1): Incorrect.
- The y - intercept of \( r\) (3) is 2 more than the y - intercept of \( g\) (1) (since \( 3=1 + 2\)): Correct.

But since the problem's full context is not provided, we can only say that to solve it, we need to:

1. For each function, determine the slope (\( m\)) and y - intercept (\( b\)) using the slope - intercept form \( y=mx + b\) (from equation, table, or graph).
2. Compare the slopes and y - intercepts as per the given statements.

Answer:

(Assuming the common problem setup with the functions as assumed above)

• The slope of function \( g \) is 2 times the slope of function \( r \).
• The slope of function \( r \) is 2 times the slope of function \( p \).
• The y - intercept of function \( p \) is less than the y - intercept of function \( r \).
• the y - intercept of \( r \) is 2 more than the y - intercept of \( g \).

(Note: The actual answer depends on the real - world definition of functions \( p\), \( g\), and \( r\) which is not fully provided in the question. The above is a sample answer based on a common problem structure.)