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Sketch the graph of the function.

$ g(t) = | 1 - 3t | $

$\begin{aligned} g(t)=|1-3 t| &=\left\{\begin{array}{ll}1-3 t & \text { if } 1-3 t \geq 0 \\ -(1-3 t) & \text { if } 1-3 t<0\end{array}\right.\\ &=\left\{\begin{array}{ll}1-3 t & \text { if } t \leq \frac{1}{3} \\ 3 t-1 & \text { if } t>\frac{1}{3}\end{array}\right.\end{aligned}$

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Jaweria S.

November 22, 2020

Missouri State University

Oregon State University

Harvey Mudd College

University of Nottingham

we want to sketch The graph of G of T and G of T is an absolute value function. So let's first start thinking about what absolute functions look like in general. What is y equals? The absolute value of X? Look like that's your basic V shaped curve that has his vertex at 00 So we're expecting to see something similar to that, but transformed in some way, perhaps shifted left or right or stretched vertically or horizontally. So this point, let's go ahead and find some ordered pairs to plot. The pivotal point for an absolute value graph is the point where you get a Y value of zero, and so that would happen at X equals or T equals 1/3. Safety is 1/3. Then we end up with one minus three times, 1/3 inside the absolute values signs. And so we just get zero from there. Let's find a couple points to the right of that and a couple points to the left of that. So suppose we pick Ah 012 and negative one. So, for zero, we end up with the absolute value of one, which is one for one we end up with the absolute value of negative to which is to for two. We end up with the absolute value of negative five, which is five. And for negative one, we end up with the absolute value for which is for so let's combine what we know about the shape of absolute value graphs with the plotting of these points and see what we get. So 1/3 0 01 12 25 negative 14 and then we go ahead and draw the lines through those points.