Question

problem 2: part a
create a graph that shows a non - proportional linear relationship with a slope of \\(\frac{1}{4}\\).

Explanation:

Step1: Recall Non - Proportional Linear Relationship

A non - proportional linear relationship has the form \(y = mx + b\) where \(b
eq0\). The slope \(m=\frac{1}{4}\). Let's choose a \(y\) - intercept \(b\) (not equal to 0). Let's take \(b = 1\) (any non - zero value would work). So the equation is \(y=\frac{1}{4}x + 1\).

Step2: Plot the y - intercept

The \(y\) - intercept is at \((0,1)\). But in the given graph, the starting point is at \((0,0)\). Wait, maybe we can adjust. Wait, the problem says "non - proportional", so the line should not pass through the origin. Let's use the slope formula \(\text{slope}=\frac{\text{rise}}{\text{run}}=\frac{1}{4}\). So from a point, we move up 1 unit and right 4 units. Let's start from a point that is not the origin. Let's say we start at \((0,2)\) (so \(b = 2\)). Then from \((0,2)\), moving up 1 (to \(y = 3\)) and right 4 (to \(x = 4\)), we get the point \((4,3)\). Then we can draw a line through \((0,2)\) and \((4,3)\) (and other points obtained by the slope) on the given grid.

Step3: Draw the Line

Using the slope \(\frac{1}{4}\) (rise 1, run 4) and a non - zero \(y\) - intercept (to ensure non - proportionality), we plot points. For example, starting at \((0,1)\), next point is \((4,2)\) (since \(x\) increases by 4, \(y\) increases by 1), then \((8,3)\) etc. We can draw a line connecting these points. The key is that the line does not pass through the origin (since proportional relationships pass through the origin \(y=mx\) with \(b = 0\)) and has a slope of \(\frac{1}{4}\).

Answer:

To create the graph:

1. Use the equation of a non - proportional linear relationship \(y=\frac{1}{4}x + b\) where \(b

eq0\) (e.g., \(y=\frac{1}{4}x + 1\)).

1. Plot the \(y\) - intercept (e.g., \((0,1)\)).
2. Use the slope \(\frac{1}{4}\) (rise 1, run 4) to find other points (e.g., from \((0,1)\), move 1 unit up and 4 units right to get \((4,2)\), then \((8,3)\) etc.).
3. Draw a line through these points. The line should not pass through the origin (to be non - proportional) and have a slope of \(\frac{1}{4}\).