Question

How do you graph `y = 4 - |x+2|`?

Answer 1

Shift |x| 4 units up, 2 units left, and have it face downwards

Explanation:

4 - |x + 2| is a constant (4) minus a some form of absolute value function. (|x + 2|).

This can be rewritten as 4 + -|x + 2|.

Since an absolute value function is always positive (by definition), and multiplying a positive by a negative always produces a negative, then we know that as x gets larger, -|x+2| gets smaller from both directions of the maximum value of the function.

There is no coefficient on the absolute value function, so we know that for every one unit right the function goes, it will go one unit up or down, depending on which side of the maximum value the x value is. If x is greater than the x value of the maximum x value, it goes down one unit for every one unit right, and the opposite for any unit to the left of the maximum x value.

Now, to find the maximum value, you find the largest value of the function 4 - |x + 2|. Well, since -|x + 2| decreases in both directions, we need to find the value for which |x+2| is the smallest, so we can subtract the smallest amount from 4.

Well, the smallest value an absolute value function can take is 0. So take what's inside the absolute value function, and solve for that (x+2) is equal to 0

x+2=0
x=-2
So the maximum x value (or the point where the function changes directions) is -2. When you input x=-2 into the function, you get y=4, so that's the maximum y value

In conclusion, you can graph the highest point of the function at (-2,4), and then go down in each direction by one unit for every one unit you move across.