Graph each function. See Examples 1 and 2.

h(x) = |-½ x|

Question

Graph each function. See Examples 1 and 2.

h(x) = |-½ x|

Accepted Answer

Welcome back. I am so glad you're here. We're asked to plot the given function. Our given function is F of X equals the absolute value of negative 7/8 X. And we're given a graph, we have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle, 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 five to positive five as well. All right. So we want to take a look at our function and ask what is its base function? What's being done to the X here? And we are taking the absolute value of X. So our base function is Y equals the absolute value of X. And we recall from previous lessons that that is going to be like a V, we have the vertex at the origin and then it's heading has one arm heading out toward positive infinity for the X and Y values and one arm heading out from the origin toward negative infinity for the X values, but positive infinity for the Y values. And those are at equal rates. So there's our Y equals the absolute value of X. What's different about our given function? Well, two things. One, this X is being multiplied by 7/8. And we look at this separately. The other is that the X is being multiplied by a negative. So what do those two things mean? First looking at this multiplication by 7/8 now this is happening directly to the X that's inside of those absolute value bars. And when that happens, we look at that number and we find out if it's between zero and one or if it's greater than 1, 7/8 is between zero and one. And because of that, it indicates horizontal and the horizontal shift is again because it's directly to the X and because it's between zero and one, it's going to be horizontal stretching, the other transformation that's taking place here, we have this negative being multiplied by our X when that's happening directly to the X, that's inside the absolute value bars that is reflection over the Y axis. So those are the two transformations taking place. So we can look at this first discussing that reflection over the Y axis. This absolute value function is already reflected over the Y axis. If we switch the arms, they're just going where the other arm was. So that's actually not gonna make a difference with the horizontal stretching. It's by a number very close to one. So it's not gonna be a very significant stretch. So it's just going to stretch out a bit and I'm just drawing this for emphasis. So it's going to look something like this still having its vertex at the origin but stretched out horizontally a little bit. But we can be more accurate with that. We can create an XY table and we can come up with some X values, plug those into our function with the absolute value of negative 7/8 X come up with the corresponding Y values, plot those points. And then we'll have our accurate final answer of a graph, right? Our domain looking at the graph is from negative infinity to positive infinity. So we could choose anything we want for our X values. I'm going to choose 01 and eight seventh. Now plugging those values in to our function plugging in zero for X, we have the absolute value of negative 7/8 multiplied by zero. Well, negative seven eights multiplied by zero is zero. So we're taking the absolute value of zero which is zero. So now plotting our final answer of a graph, we can clear these others and we'll have one point at the origin. All right, now let's use one for X. The absolute value of negative 7/8 multiplied by one. Well, negative 7/8 multiplied by one is negative 7/8. So we're taking the absolute value of negative 7/8 which is a positive 7/8. So from the origin, we are going to the right one and then we're going up 7/8. So almost one. Alright, last X value we have negative, the absolute value of negative 7/8 multiplied by eight seventh, negative 7/8 multiplied by eight seventh is a negative one. So we're taking the absolute value of negative one which is a positive one. And from the origin, we'll go to the right eight seventh which is just past one and then we'll go up one. All right. So we know that the same thing is happening on both sides of the Y axis. We've got symmetry. And so we can draw our point from the origin passing through these two points that we plotted heading toward positive infinity for the X and Y values. And then on the other side to the left of the Y axis, we can put mirrored points. We'll have a negative one, 7/8 and a negative eight seventh one for two points. And we can draw the line from the origin heading up through those two points toward negative infinity for the X values and positive infinity for the Y values. And that is our F of X equals the absolute value of a negative 7/8 X. Well done. We'll catch you on the next one.

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Graph each function. See Examples 1 and 2. h(x) = |-½ x|