Graph each function. See Examples 1 and 2.

ƒ(x) = -3|x|

Question

Graph each function. See Examples 1 and 2.

ƒ(x) = -3|x|

Accepted Answer

Welcome back. I am so glad you're here. We're asked to plot the given function. Our given function is G of X equals negative 15, multiplied by the absolute value of X. Then we're given a blank graph. We have a vertical Y axis and a horizontal X axis. They come together at the origin. The range for what's drawn for our Y axis is from negative to positive 10. And the domain for what's drawn for our X axis is from negative five to positive five. All right. So we take a look at our given function and we ask ourselves, what is its base function? What's happening to the X here? And we're 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 Y equals the absolute value of X is a function that essentially looks like a V, it has its vertex at the origin. And then it has one arm heading up into the second quadrant toward positive infinity for Y and negative infinity for X at the same rate. And then the other arm is also starting at the origin. But heading up into the first quadrant for positive infinity for both X and Y also at the same rate. And there's our Y equals the absolute value of X function. So then we look at our given function G of X and ask what's different. There are two differences here. One is that the absolute value of X is being multiplied by a 15. And then we look at this separately, it's being multiplied by a negative. What do each of those mean? Well, because we're multiplying that 15 by the absolute value of X, it's not directly to that X, it's not inside the absolute value bars with it because it's outside of those absolute value bars, it indicates a vertical change. And because the number 15 is larger than one, it indicates vertical stretching. Now multiplying this negative here, that's also outside of the absolute value bars. So because we're multiplying by that negative, not directly to the X, that indicates reflection over the X axis. So looking back at our Y equals the absolute value of X, we can see that that's going to flip over the X axis. So it's going to open downward and then it has vertical stretching. So we can draw arms going toward negative infinity for both X and Y has its vertex at the origin and then an arm going toward negative infinity for Y but positive infinity for X. Now that gives us an approximate idea of what our function is doing, but we can be a little bit more specific because we know generally what's going on, we can plot just a couple points to get it more accurate to do that. We can create an XY table. We'll choose a few values for X looking at our function, seeing what the domain is, we plug that into our function negative 15 multiplied by the absolute value of X discern the appropriate Y values and then plot those points. So for the X values here, we can see looking at the graph that our domain is from negative infinity to positive infinity. So I'm going to choose X values of negative 20 and positive two. Now plugging those into our function negative multiplied by the absolute value of not X but negative two. Well, the absolute value of negative two is positive two. So we have negative 15 multiplied by two which is negative 30. Now we can clear out these graphs and plot our final answer here for the point negative two negative 30 from the origin, we go to the left two and down 30. All right. Now for an X value of zero, negative 15 multiplied by the absolute value of zero. Well, the absolute value of zero is zero. So negative 15 multiplied by zero is zero. There's our point at the origin. Now for an X value of positive two, negative 15 multiplied by the absolute value of two. Well, the absolute value of two is two. So we have negative 15 multiplied by two again, which is still negative 30 plotting the 0. negative 30 from the origin, we go to the right two and down 30. And now we can draw our lines. So we have our vertex at the origin. It's opening downward, one line begins at the origin and passes through negative two, negative 30 heading toward negative infinity for both X and Y. And the other point begins at the origin passes through our 0. negative 30 heading toward negative infinity for the Y values and positive infinity for the X values. And there is our G of X which equals negative 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

Transformation of Functions

Graphing Techniques

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