Question

function a and function b are linear functions.
function a

x	y
-4	-2
-1	-2
5	-2

function b (graph)
function a and function b have different rates of change. using only integers, write the equation of another function whose slope is between those two rates of change.
y = \boxed{}x + \boxed{}

Explanation:

Step1: Find slope of Function A

For Function A, using points \((-4, -2)\), \((-1, -2)\), \((5, -2)\). The slope \(m_A=\frac{y_2 - y_1}{x_2 - x_1}\). Taking \((-4, -2)\) and \((-1, -2)\), \(m_A=\frac{-2 - (-2)}{-1 - (-4)}=\frac{0}{3}=0\).

Step2: Find slope of Function B

For Function B, from the graph, two points: when \(x = 0\), \(y = 5\) (y - intercept) and when \(x=-2\), \(y = - 5\) (approx, or use \(x = 1\), \(y = 10 - 5?\) Wait, better: take two points on the line. Let's take \((0,5)\) and \((-2, -5)\). Slope \(m_B=\frac{-5 - 5}{-2 - 0}=\frac{-10}{-2}=5\)? Wait, no, looking at the graph, when \(x = 1\), \(y\) is 10? Wait, the line passes through (0,5) and (1,10)? Wait, no, the grid: each square is 1 unit. Let's take two points: (0,5) and (-1,0)? No, better: the line goes from ( - 2, - 5) to (0,5). So \(m_B=\frac{5 - (-5)}{0 - (-2)}=\frac{10}{2}=5\). Wait, but let's check another point. If \(x = 1\), \(y = 10\), so from (0,5) to (1,10), slope is \(\frac{10 - 5}{1 - 0}=5\). So \(m_B = 5\), \(m_A=0\).

Step3: Choose a slope between 0 and 5, say 3. Then choose a y - intercept, say 0 (or any integer). So the equation can be \(y = 3x+0\) (or \(y = 2x + 1\), etc. Let's take slope 3, y - intercept 0. So \(y = 3x+0\) (or any slope between 0 and 5, integer, and any integer y - intercept).

Answer:

\(y = \boxed{3}x+\boxed{0}\) (Note: Other valid answers like \(y = 2x+1\), \(y = 4x - 2\) etc. are also correct as long as slope is between 0 and 5 (integer) and y - intercept is integer.)