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a. Sketch the graph of $y=|x| .$b. Sketch the graph of $y=-|x| .$c. Describe the graph of $y=-|x|$ in terms of the graph of $y=|x|$

a) see graphb) see graphc) The graph $y=-|x|$ is the graph of $y=|x|$ reflected about the $x$ -axis.

Algebra

Chapter 4

RELATIONS AND FUNCTIONS

Section 4

Absolute Value Functions

An Introduction to Geometry

Functions

Linear Functions

Polynomials

Missouri State University

Oregon State University

Harvey Mudd College

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absolute value. Functions are parent function is f of X equals the absolute value of X. And that is because, um it does not have a horizontal shift, which means the Vertex starts at X equals zero. It does not have a vertical shift on the outside, which means the Y value is also zero. So this has the Vertex at 00 and the slope value is a positive one. So that means it's going to open up and we'll take a look at that. Well, here's the graph of that function. Where the Vertex is at 00 to the right of the graph are absolute value. Our slope is positive. One to the left of the graph. We used the opposite slope, so that's negative. One. How does that compare to a part of the family? Absolute value. Negative, absolute value of X. So again, there is no horizontal shift inside the absolute value bars. There's no vertical shift to the right of the absent value bars, so the Vertex is still at 00 but we now have instead of a is equal toe. One. We now have a is equal to negative one, which means we have the opposite slope. So to the right of the Vertex, we have a negative one slope and to the left of the Vertex we have opposite of that which would be a positive one slope. So this should open down because we have the opposite slopes. So let's take a look at it and verify that a negative sign to the left of the Absolute Value bar states that your avert your absolute value graph is going to open down. So I've got the absolute value of X O. But I need to go and put a negative sign in front of that. So let's put a negative, sign him from that and see what that looks like. So it agrees with our discussion that the Vertex is still a 00 There is no horizontal, a vertical shift, and we are using the opposite slope to the right of the Vertex. Instead of a positive one soap. We have a negative one soap to the left of the Vertex. We used the opposite of the slope that you see, and so that gives us a positive. So again, a negative sign to the left of the absolute value bars. Some people will say it flips it upside down, but the proper vocabulary is the absolute value graph opens down.

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