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### Course: Algebra 1>Unit 10

Lesson 1: 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.
• 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.
• 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|.
• I really want to know where this function is used in real life
(1 vote)
• Tell me. When you do, for example, pushups, are you practicing being able to get up off the ground or push a bunch of rubble off your back like spider-man? No, you're trying to gain strength in certain muscle groups that will allow you to do harder things later. Same thing goes for this kind of math. Usually the things you learn here are to prepare you for harder math. It's a way to solidify the foundations for the next bricks of learning. It's to help you think in a new way that will make future math easier and will make it more intuitive.
Hope this helps.
• Why does he shift to the left instead of the right?
• The standard absolute value graph y=|x| has its vertex at (0, 0). If you want to change the point to be at (3,0), that means you are making x=3. Notice, these are on opposite sides of the "=". if you need to place them on the same side of the "=", then you would have x-3=0. This is essentially what is being done in the function. The X and 3 are are on the same side of the equation. So, we use y=|x-3| to shift to x=3.

By contrast, if you want to shift the vertex up/down, the "y" and the value to move are on opposite sides of the equation. So, we can use the same sign we want. y=|x|+4 moves the graph up 4, and y=|x|-4 moves the graph down 4.

Hope this isn't too confusing. It may help if you try experimenting with the equation to see how it works.
• 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.
• How do I know how much I move the slope when graphing?