Okay. So let's give this problem a try. Here we have the function \( f(x) = c \times x^2 \), and we're asked to graph our function \( f(x) \) when \( c = 2 \) and when \( c = 0.5 \). Now what we have on the graph already here is a curve which represents our function \( f(x) = x^2 \). This is the general case for a parabola, but we need to figure out what's going to happen if we take various constants and plug them into this function. So let's see. We're going to start for the case where our constant here is equal to 2. This means that \( f(x) = 2 \times x^2 \). This is what the function is going to look like if we plug 2 into the constant. Now what I'm going to do is try a bunch of different \( x \) values. I'll first try an \( x \) value of 0. If this were to happen, we would have \( 2 \times 0^2 \), and \( 2 \times 0 \) is 0. \( 0^2 \) is just 0. So that means we're going to have a point at 0. That's going to be one point we can plot here. Now what I'm also going to do is plot a value of -1. If I do this, we're going to get \( 2 \times (-1)^2 \), just replacing this \( x \) with -1. In this case, \( 2 \times (-1) \) is -2, so we'll have -2 squared, and -2 squared is actually positive 4. So that means that a value of -1, I mean, at -1, we're going to be at a \( y \) value of 4. So this is another point that we could put on this graph.
Now lastly, I'm going to try a point of positive 1. So in this case, we'll have \( 2 \times 1^2 \). \( 2 \times 1 \) is 2, and 2 squared is 4. So at a value of 1, we're going to or at an \( x \) value of 1, we're going to have a \( y \) value of 4. So our graph is going to look something like this when we replace this constant that we see here with a 2. And I think this actually makes sense because recall in the previous video, we discussed whenever you have a constant multiplied inside of your function, it's going to cause a horizontal stretch or shrink to your graph. In this case, we saw that it caused a horizontal shrink, and this happens whenever your constant is greater than 1. So it makes sense that we would get this situation where the graph shrinks since 2 is greater than 1. But now let's try this other situation where we have \( c = 0.5 \). This basically means our constant is between 0 and 1.
So let's see what happens if I do this. Well, in this case, our function \( f(x) \) is going to become \( \frac{1}{2} x^2 \), because now we're just gonna take this constant and replace it with the \( \frac{1}{2} \) that we have over here. Let's see how this behaves. What I'm first going to do is plug in an \( x \) value of 0 like we did before, in which case we have \( \frac{1}{2} \times 0^2 \), and anything multiplied by 0 is just 0, so we already know this whole thing will come out to 0, meaning we'll have the same origin point 0. Now next, what I'm going to do is I'm actually going to try an \( x \) value of 2. And the reason that I'm trying 2 specifically is that if I take this \( x \) and replace it with a 2, notice we're going to get \( \frac{1}{2} \times 2^2 \). And \( \frac{1}{2} \) and 2 will actually cancel each other here because taking 2 and cutting it in half will just give you 1. So really you're just getting up with 1 squared, which is just 1. So if you try an \( x \) value of positive 2, you're going to end up here at a \( y \) value of 1. And likewise, if you were to try -2, well, in this case, we would have \( \frac{1}{2} \times (-2)^2 \), in which case this 2 would cancel with that one giving us -1 squared. And -1 squared is just positive 1 because -1 times -1 will cause the negative signs to cancel. So at an \( x \) value of -2, we're going to be at a \( y \) value of positive one again, meaning our function is going to look something like this when we multiply the inside by \( \frac{1}{2} \). And notice in this case, we got a horizontal stretch because whenever our constant is between 0 and 1, we get a horizontal stretch, whereas when our constant is greater than 1, we get a horizontal shrink.
Now one more thing I want to mention before finishing this video is notice that when we had the horizontal shrink, it almost appeared like we had a vertical stretch. Because in this situation where we had the horizontal shrink, it looked like vertically the graph almost stretched. And likewise, when we had a horizontal stretch, this kind of looked like we had a situation with a vertical compression or shrink. And that will often be the case when you see these types of graphs that have symmetry on them, is that the horizontal stretches will often be similar to the vertical shrinks, and the horizontal shrinks will be similar to the vertical stretches. So that's just something to keep in mind visually when looking at these graphs. But either way, this is how you solve the problem, and these are the answers. So hopefully, you found this helpful. This is what the graph is going to look like.