# How do you graph y= 1/3 | x-3 | + 4?

Jan 27, 2018

graph{(1/3)abs(x-3)+4 [-3.865, 11.935, -0.68, 7.22]}

#### Explanation:

Let's start with the function $y = x$ and describe the transformations taken in order to make the current function.

$y = x$
graph{y=x [-10, 10, -5, 5]}

First, we're taking the absolute value, meaning every negative $y$ value is flipped across the $x$-axis and made positive.

$y = \left\mid x \right\mid$

graph{y=absx [-10,10,-5,5]}

Now, we have the function in terms of:

$y = a \left\mid x - h \right\mid + k$

where $a = \frac{1}{3}$, $h = 3$, $k = 4$. So let me explain what each of these means.

The parameter $a$ is being multiplied by the $x$ values, which determines the slope of the lines, which is the $\text{rise"/"run}$ of the function, or (∆y)/(∆x). Since this is $\frac{1}{3}$, we know that for every one increase in y, we get three increases in x.

$y = \frac{1}{3} \left\mid x \right\mid$
graph{1/3absx}

Next, the h value determines how far right we shift the function. NOTE: By default, the value is negative. If there is. a plus sign, you shift this function left.

Since this value is 3, we shift this three units right.

$y = \frac{1}{3} \left\mid x - 3 \right\mid$
graph{y=1/3abs(x-3)}

Finally, we have the lonely $k$ value, which just tells us how far we shift this up. This value is four, and therefore we shift the graph 4 units up.

$y = \frac{1}{3} \left\mid x - 3 \right\mid + 4$
graph{y=1/3abs(x-3)+4 [-3,11,-1,8]}