Graph each function. See Examples 6–8.g(x) = |x| - 1

Question

Accepted Answer

Welcome back. I am so glad you're here. We're asked to draw the graph of the following function. Our function is F of X equals the absolute value of X minus four. Then we have 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 20 to positive 20. And the range for what's drawn for our YX axis is from negative 20 to positive 20 as well. All right. So we want to take a look at our given function and figure out what is its base function. So when we look at this, we see that what's happening to our X is that we're taking the absolute value of it. So our base function is Y equals the absolute value of X. So we can draw that on our graph. We recall from previous lessons that, that looks like a V, it's got its vertex at the origin and then it has one arm heading toward negative infinity for the X values and positive infinity for the Y values. And that's in the second quadrant and then the other arm looks just like it in the first quadrant heading toward positive infinity for both the X and Y values. So there's our Y equals the absolute value of X. Now, what's different in our given function? The only difference is that in our F of X, after we take the absolute value of X, we're subtracting four. Now, that is not happening directly to the X, it's not inside of the absolute value bars. It's not absolute value of X minus four. The subtracting of four comes after we've taken the absolute value. And because it's not directly happening to that X, that indicates a vertical shift. And more specifically because we are subtracting four, the vertical shift is down four units. And with this one, we know that our vertex is the easiest part to identify that's going down four units. And we can draw that with the vertex at negative four for the Y value. So it would be zero, negative four. But we can be more specific about this as well. By plotting a few points, we can draw an XY table, choose a few values for X plug those in to the absolute value of X minus four, get the associated Y values and then plot those points. And we know generally what's going on with this function. So we only have to choose a few X values. And we, we have a pretty good idea of what's going on. So for the X values. I'm just going to choose five zero and negative five. And now plugging these in for X for a value of five for X, we have the absolute value of five instead of the absolute value of X. The absolute value of five is five and five minus four is a positive one. So clearing our graph now, we'll have our final answer in here. We've got from the origin, we'll go to the right five units and we'll go up one. Now let's use zero for X. We have the absolute value of not X but zero minus four. The absolute value of zero is zero and zero minus four is negative four from the origin. We'll go down four units. All right, the last X value negative five, put the absolute value of not X but negative five minus four. The absolute value of negative five is positive five. So we have five minus four, which is again a positive one from the origin. We'll go to the left five and up one. So now we can connect these points understanding that we have one arm heading toward negative infinity for the X values and positive infinity for the Y values. The vertex is at zero, negative four and one arm is heading toward positive infinity for both the X and Y values. And there's our graph well done. We'll catch you on the next one.

Graph each function. See Examples 6–8.g(x) = |x| - 1

Key Concepts

Absolute Value Function

Vertical Shifts

Graphing Techniques

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