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## Get ready for Algebra 2

### Course: Get ready for Algebra 2>Unit 3

Lesson 7: Graphs of absolute value functions

# 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).

## Want to join the conversation?

• 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?
• The "V" will point down instead of up because of the negative sign.
• 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.
• what if we're taking the absolute value of 2-x?
• Let's try to simplify it. We have a function f(x) = |2-x|. that is the same thing as |-x+2|. Now we multiply by -1 inside the ||. That is totally legitimate because, well, it's a absolute value :) . So f(x) = |x-2|.
• 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.
• In this video the teacher says when x is less than -3, it makes the value negative, and the absolute value positive, and that's why it slopes down. And when above -3 it makes the value positive, and that's why it slopes up.
But the line does not slope down or up because of a positive or negative value inside the absolute value lines, it's because the absolute value itself is always positive in this scenario.

The way the teacher words things is so confusing sometimes and I sit here for an hour trying to re-teach myself what he could possibly mean when he says things like this.

The value inside the absolute value doesn't determine the direction of the line, the positive or negative of the absolute value determines in. So if y=-|x+3|, the line to the left of -3 would slope up. Right? Because the y value would then be less than 0.
• Absolute value functions create V-shaped graphs.
The minus in front of your absolute value tells you that the V-shape will open downword. So, the line to the left of -3 will slope upward toward x=-3. And it will slope down as it moves to the right of -3.
• 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.
• He's using each color to represent a different equation and its graph. The red graph is the equation: y=|x+3|. The orange graph is the equation: y=2|x+3|.
Hope this helps.
• 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 do I graph f(x)=-|x|
(1 vote)
• with absolute value, the positive domain of x would be the same as f(x) = x and the negative domain of x would be the same as f(x) = - x. This creates the V shape of absolute value.