How do you graph #y<=3absx-1#?

1 Answer
Feb 5, 2018

See answer below.


Let's build it up step by step. Firstly, we will ignore the inequality and start with the simplest linear relationship

graph{x [-10, 10, -5, 5]}
To get the equivalent absolute value function, simply make all the negative x-values positive. Notice that the function is reflected in the y-axis. One way of drawing it is to just draw the right hand side of the y-axis (positive x-values) and reflect it in the y-axis.

graph{|x| [-10, 10, -5, 5]}

The first transformation to make to the original graph is to change the gradient to 3. This makes the graph steeper.

graph{3x [-10, 10, -5, 5]}
To draw the absolute version, look at #y=3x# and just draw the right half (positive x-values), then reflect it in the y-axis.

graph{3|x| [-10, 10, -5, 5]}
The second transformation is a vertical translation of -1. This just means that you shift the whole graph down by 1 unit. Hopefully you can see that the graph of #y=3x# is now just moved down 1 unit.

graph{3x-1 [-10, 10, -5, 5]}
Again, just draw the right half of #y=3x-1# (positive x-values) then reflect it in the y-axis to get the absolute graph

graph{3|x|-1 [-10, 10, -5, 5]}
Lastly, we have to deal with the inequality. It tells us that the required y-values must be less than or equal to the line of the function. We shade the required region, which means that we want to shade the whole region under the line. Because it is less than or equal to, we use a solid line for the graph. If it was strictly just less than, we would draw a dotted line for the graph. The final answer looks like this