Graph each function. See Examples 1 and 2.

ƒ(x) = ⅔ |x|

Question

Graph each function. See Examples 1 and 2.

ƒ(x) = ⅔ |x|

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 7/8 multiplied by the absolute value of X. And then we're given a blank graph. We have a vertical Y axis, a horizontal X axis and they come together at the origin in the middle, the domain for what's shown 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 five to positive five as well. All right. So we want to take a look at our function and ask ourselves what is the base function? What's happening to our X? And we are taking the absolute value of X. So our base function is Y equals the absolute value of X. We recall from previous lessons that that function is shaped like a V, it has a vertex at the origin and it's opening upward. So we're going to have one arm coming from the origin and heading up toward positive infinity for Y and negative infinity for X in the second quadrant and going in between those two axes at the same rate, then we have another arm reflected over the Y axis again, from the origin heading up toward positive infinity for both X and Y at the same rate. And there's our Y equals the absolute value of X. Now, what's the difference between that and our given G of X? Well, G of X is being multiplied it, it has this absolute value of X being multiplied by 7/8. And because that 7/8 is not being multiplied directly to the X, that's after the absolute value has been taken, that indicates a vertical shift. And more spec specifically because 7/8 is in between zero and one, it indicates vertical shrinking. Vertical shrinking is going to take us slightly farther away from the Y axis. So we still have our vertex at the origin. We're still heading in those same directions in the 1st and 2nd quadrant, but it's gonna look a little bit lower than our Y equals the absolute value of X. So we start at the origin and we head toward negative infinity for the X values just slightly faster than positive infinity for the Y values in the second quadrant. And then reflecting that in the first quadrant again, we're starting at the origin heading toward positive infinity for X slightly faster than we're heading toward positive infinity for Y. But that's approximate, that's not exact, it's actually pretty darn close, but we can be more accurate by getting an XY table, choosing some X values, plugging those into our function multiplying 7/8 by the absolute value of X. And then we'll figure out those Y values and plot those exact points. So we look at our graph and we see that our domain for X is from negative infinity to positive infinity. But if we go by what's shown on here, we can keep our X values within what's drawn. And I'm going to choose X values of negative 10 and positive one. Now we can plug those into our function. We have 7/8 multiplied not by the absolute value of X, but by the absolute value of negative one and the absolute value of negative one is positive one. So we have 7/8 multiplied by one which is 7/8. Now, if we clear out these other graphs that helped us do our work, now we can create a final draft of that from the origin. We move to the left one space one unit and then up 7/8 of the unit, which is just under one for our first point negative 1, 7/8. Now we'll figure out what our Y value is. When we have an X value of zero, we have 7/8 multiplied not by the absolute value of X, but by the absolute value of zero. And the absolute value of zero is zero. So we have 7/8 multiplied by zero, which is zero. And there's our point at the origin. Last value for X is one, we have 7/8 multiplied by the absolute value of not X but one, the absolute value of one is one. So we have 7/8 multiplied by one again, which is still 7/8 from the origin, we go to the right one and up 7/8 of a unit. Now we have three points, we know generally what this graph looks like. So we can finish drawing it and connect our points from the origin heading into the second quadrant. We will be passing through the point negative 1 7/8 heading toward negative infinity for the X values and positive infinity for the Y values. And then also from the origin, we're heading into the first quadrant passing through the 0.1 7/8 and heading toward positive infinity for both the X and Y values. And there is our graph G of X equals 7/ 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) = ⅔ |x|

Key Concepts

Absolute Value Function

Vertical Scaling

Graphing Techniques

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