Graph each function. See Examples 1 and 2.

g(x) = ½ x²

Question

Graph each function. See Examples 1 and 2.

g(x) = ½ x²

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 1/12 X squared. And then we are given a blank 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 0.1 to positive 0.5. All right. So taking a look at our function, we want to ask ourselves what is our base function? What's happening to the X here? And the X is being squared. So our base function is Y equals X squared. We recall from previous lessons that Y equals X squared is an upward facing parabola. Now, with the scale that we have for this graph, it's not going to look like the typical parabola. So I'm I'm going to draw it as though the scale for the Y axis is the same as the scale for the X axis. So sort of ignoring the labels for our Y axis for a moment here. So for our upward facing Parabola, it's heading up toward positive infinity for the Y axis and negative infinity for the X axis in the second quarter, second quadrant, excuse me. And then it curves down and passes through the origin, which is the vertex where it turns around, heads back up toward both positive infinity for the Y axis and positive infinity for the X axis. And so that's our Y equals X squared. Now, we ask ourselves, what is the difference between that and our G of X? And the difference here is that our X squared in G of X is being multiplied by 1/12. So that's being multiplied after the X gets squared. It's not happening first and directly to our X. So because that's happening after the X is squared, that indicates a vertical change and specifically because 1/12 is between zero and one, it indicates vertical shrinking. Now, from our Y equals X squared. If we were to indicate vertical shrinking, then again, we're ignoring the Y axis uh labels for, for a moment, it's going to look like it's lower for vertical shrinking. So it's our same upward opening parabola, but it's going to look as though it's a bit lower. So that's the transformation that's occurring. But this one isn't accurate because we have this very zoomed in Y axis. So we can make this accurate. The way to do that is to create an XY table. We're going to choose some X values, plug those into our function 1/12 multiplied by X squared. And then we figure out what the Y values are and plot our points. So we can see from our graph that the domain for our X axis is from negative infinity to positive infinity. We can choose any X values we want. But here focusing on what's drawn, I'm going to choose X values of negative 20 and positive two. So let's plug negative two in. For X, we have 1/12 multiplied not by X squared, but by negative two squared, negative two squared is positive four. So we have 1/12 multiplied by four, which is a positive one third. So now we can plot that point clearing our graph, the point negative 21 3rd from the origin. We go to the left two and then we're going up one third which is just past 0.3. All right, we've got one point. And so next, we'll have 1 12 multiplied by not X but zero squared, zero squared is zero. So we have 1/12 multiplied by zero, which is zero. There's our point at the origin, right. The last X value is two, we have 1/12 multiplied not by X squared, but by two squared, two squared is four. So we have 1 12 multiplied by four, which is still one third from the origin. We go to the right two units and up one third. All right, we can connect these lines or these points as smoothly as possible. Again, we're heading up toward positive infinity for the Y values and negative infinity for the X values in the second quadrant coming down through our point negative 21 3rd. Then curving, when we get close to the origin, passing through that curving back upward, passing through our 0.21 3rd and heading toward positive infinity for both the X and Y values. And you can see if we would have plotted some points for our Y equals X squared. That one's going to look really skinny. With this scale, it would be very close to the Y axis if we would have been plotting that Y equals X squared. But with the vertical shrinking with this scale, our graph of G of X equals 1/12 X squared looks like a normal parabola. Alright, there's our graph well done. We'll catch you on the next one.

Graph each function. See Examples 1 and 2. g(x) = ½ x²

Key Concepts

Quadratic Functions

Graphing Techniques

Transformation of Functions

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