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# Graphing absolute value functions

We can graph any absolute value equation of the form y=k|x-a|+h by thinking about function transformations (horizontal shifts, vertical shifts, reflections, and scalings).

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• What if there is a number in front of X?(inside the absolute value bars) • if f(x)=|x+3|, we know that the graph need be shifted 3 units to the left of the origin. this was obtained by equating x+3 to 0, which gives us x= -3. plugging x=-3 in f(x),
f(x)=|-3+3|=0.
this is the vertex of the graph; the point(-3,0) at which the value of y is least.
if our f(x), for example, were to equal |2x+3|,
doing 2x+3=0 would give x=-3/2.
now, the vertex of the graph of our new f(x) would be (-3/2,0)
• What if the +2 outside of the absolute value bars is a -2? • • I am not understanding how to do stretching by factors • As I've posted before many times, have you done linear equations before?

In y = 2x, the slope is 2. For every x, we get a y that is twice as large. This causes the line to be quite steep; having a fraction makes the line less steep.

Same thing here. y = 2|x| means that for every x, y is twice as large. The V-shape is compressed. However, y=1/2|x| would be stretched out.
• Why does the graph get shifted to the left if the value inside the absolute value brackets is positive like if the equation is:

Y = |x+4|

Why does the function get shifted four to the left? Doesn’t that seem a bit counterintuitive and wrong? I know Sal has explained this multiple times in different videos. But i just don’t understand why. Could someone please explain this to me? • There are several ways I look at it. Lets do y=|x+4|+3. We can see this moves 4 units to the left and 3 up. Let's subtract 3 to get y-3=|x+4| which is the equivalent equation. Note that now both x and y do the opposite of what we might "expect," y is 3 units up and x is 4 units to the left. The other idea is to think about it like "moving the zero" even though we do not actually do this. If we choose x=4, we end up moving the 0 4+4=8 units, but if we choose x=-4, we end up with the new -4+4=0. Think also of a circle (x-h)^2+(y-k)^2=r^2, in both cases, you have to change to signs of x and y to "move the 0." You are not the first to see it as counterintuitive, but that should not lead to the conclusion that it is wrong. This may not be very satisfying to you.
• why did he move from the red to the orange color? I understand what he did, but the why is confusing and doesn't make sense. Get back soon. Thanks. • What do you do if there is a negative fraction in the absolute value. ex. |-1/2x + 3| -10?
(1 vote) • I'll assume you're asking how to graph the equation. If you have a coefficient of x inside the absolute value sign, one thing you can do is try and isolate it a little bit, by setting it as a factor to the rest of the inside of the absolute value. If you do that to this problem, you'll get this:
y = |(-1/2)(x - 6)| - 10
Now you're taking the absolute value of something (x - 6) times a negative. Because absolute value doesn't care about the sign, you can effectively just remove the negative on the 1/2. Now that the equation has been simplified to y = |1/2 (x - 6)| - 10, you can get to graphing.
For any function, if you have a coefficient inside the operation of the function (the absolute value bars in this case), it basically does the opposite of a coefficient on the outside. While a high coefficient on the outside would increase every y-value by a certain factor (vertically stretch the graph), a high coefficient on the inside would increase every x-value by a certtain factor (horizontally stretch, which makes the graph wider). Since we have a low (<1) coefficient inside the function, the graph will horizontally get squished, or vertically stretch. So our correct graph should be less steep than a normal absolute value function, and translated down and to the right.
To actually put numbers onto this, you would lookk at your equation to get the vertex of the function (6, -10) and use the positive and negative slope to draw the two parts of the function.
• • How can you tell where the X or Y is increasing or decreasing? • In general, you only care about the dependent variable increasing or decreasing ("y" in this case). It doesn't exactly make sense to think about the independent variable ("x") to be increasing or decreasing.

To be able to tell if "y" is increasing or decreasing, you should look at the graph. Read the graph from left-to-right. When it goes down, it's "decreasing", when it goes up, it's "increasing".

If you don't have access to the graph, look at the coefficient on "x" (look at the number to the left of the absolute value symbols). If that number is positive, then it opens up, and to the left of the vertex is decreasing, and to the right is increasing. If the coefficient is negative, it opens down, and the to the left of the vertex is increasing, and to the right is decreasing. 