Graph each function. See Examples 1 and 2.ƒ(x) = 3|x|

Question

Accepted Answer

Welcome back. I am so glad you're here. We are asked to plot the given function. Our given function is G of X equals 15, multiplied by the absolute value of X. Then we are given a graph, we have a vertical Y axis and a horizontal X axis. They come together at the origin. The domain for what's drawn for our X axis is from negative five to positive five. And the range for what's drawn for our Y axis is from negative to positive 80. All right. So the first thing we want to do is ask ourselves, what is our base function? What is happening to our X? And we're taking the absolute value of it. So our base function is Y equals the absolute value of X. And we recall from previous lessons that, that essentially looks like a V with its vertex at the origin and its opening upward. Now this graph is scaled very zoomed out for our Y axis. So when we draw this, it's going to look like a little bit of a smushed V and it's not going to be exact, but just to give you an idea because we'll have points going through the origin and then we'll go through negative 11 and positive 11. So from the origin, we'll have one arm of our V heading toward negative infinity for the X values and positive infinity for the Y values. It's heading out at the same rate between those two axes, but these are scaled differently. So it looks like it's very wide. And then we have that reflected over the Y axis from the origin into the first quadrant heading through the 0.11 and heading toward positive infinity for the X and Y values at the same rate, there's our Y equals the absolute value of X. Now, what's the difference between that and our given function G of X? Well, in G of X, we're taking our absolute value of X and multiplying it by 15, we're multiplying it by 15 after we've taken the absolute value. So that's not happening directly to the X. And so that means that it's going to be a vertical shift. And more specifically because 15 is greater than one, it indicates vertical stretching and vertical stretching is going to move us closer to the Y axis. It's going to do so by a factor of 15. So just to show you generally what's happening, we still have our V shape, it still has its vertex at the origin and it's still opening upward. But now the arms are going to be closer to the Y axis. And with this scale for our y axis, it's gonna look a little bit more like a normal V. So from the origin, we have one arm heading toward negative infinity for the X values and positive infinity for the Y values. And then we have that reflected over the y axis from the origin and arm reaching toward positive infinity for both the X and Y values. And we can see how this is closer to the Y axis. But we can be more accurate than this. We can do. So by creating an XY table, we choose some values for X based on what we've drawn. And then we plug those in to our function 15 multiplied by the absolute value of X come up with the applicable Y values and plot those points. So for our X values, we can see from our graph that the domain is from negative infinity to positive infinity. But we can see from what's drawn that it would help to pick numbers between negative five and positive five. So for this one, I am going to choose X values of negative one, zero and positive one. Now we can plug those values for X into a function. For negative one, we have 15 multiplied by the absolute value of not X but negative one and the absolute value of negative one is positive one. So we have 15 multiplied by one which is 15. Now clearing away these other graphs. We can plot our final answer the first point negative 15 from the origin, we go to the left one and we go up 15. All right. Now, an X value of zero, we have 15 multiplied by the absolute value of not X but zero. The absolute value of zero is zero. So we have 15 multiplied by zero, which is zero. There's our point at the origin. Now, the last X value of one, we have 15 multiplied by the absolute value of not X but one, the absolute value of one is one. So we have 15 multiplied by one again, which is still 15 from the origin. We go to the right one and up 15. All right, we know this is going to be a V shape. So from the vertex at the origin, we'll pass through our point negative 1 15 heading up toward positive infinity for the Y values and negative infinity for the X values reflecting that over the Y axis from the origin will pass through the 0. 15 heading up toward positive infinity for both the X and the Y values. And that is our G of X equals 15, multiplied by the absolute value of X. Well done. We'll catch you on the next one.

Graph each function. See Examples 1 and 2.ƒ(x) = 3|x|

Key Concepts

Absolute Value Function

Vertical Stretch

Graphing Techniques

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