Question

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

Answer 1

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=abs(x)`

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

Now, we have the function in terms of:

`y=aabs(x-h)+k`

where `a=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 `"rise"/"run"` of the function, or `(∆y)/(∆x)`. Since this is `1/3`, we know that for every one increase in y, we get three increases in x.

`y=1/3abs(x)`
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=1/3abs(x-3)`
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=1/3abs(x-3)+4`
graph{y=1/3abs(x-3)+4 [-3,11,-1,8]}