Video Transcript
Which of the following graphs represents the equation 𝑦 equals negative 𝑥 squared?
And then we have five graphs to choose from. Now, in fact, there are a couple of ways that we can answer this question. One technique is to use a table of values. Alternatively, we can look at the equation given and see if we can get there using a series of transformations. We’re going to use the latter method and then check it with the former. That is, we’re going to begin with the graph of the equation 𝑦 equals 𝑥 squared.
The graph of 𝑦 equals 𝑥 squared is a parabola. Now, it’s a parabola that opens upwards like this, and it passes through the point zero, zero. Remember, parabolas by their definition are also symmetrical. In fact, the parabola that’s given by a quadratic function has a line of symmetry which passes through its turning point. In particular, for the function 𝑦 equals 𝑥 squared, it has a vertical line of symmetry given by the 𝑦-axis or 𝑥 equals zero.
Now, if we look at our graphs (A) through (E), we see the graph that represents 𝑦 equals 𝑥 squared must be graph (B). It’s a parabola that opens upward, it has a line of symmetry given by 𝑥 equals zero, and it passes through the point zero, zero. We can confirm this by checking a couple of points that lie on the line, for instance, the point that passes through two, four. This tells us that when 𝑥 is equal to two, 𝑦 must be equal to four. Well, if we substitute 𝑥 equals two into the equation 𝑦 equals 𝑥 squared, we get 𝑦 equals two squared, which is four, as required. We could, if we so required, do this with a couple more points, for instance, the point one, one or the point negative two, four. However, let’s move on and identify the transformation that maps the graph of 𝑦 equals 𝑥 squared onto the graph of 𝑦 equals negative 𝑥 squared.
Going back to the general form 𝑦 equals 𝑓 of 𝑥, we know we can map that onto the graph of 𝑦 equals negative 𝑓 of 𝑥 by a single reflection in the 𝑥-axis or the line 𝑦 equals zero. That means we can map 𝑦 equals 𝑥 squared onto 𝑦 equals negative 𝑥 squared by a reflection in this line. Going back to graph (B), for the equation 𝑦 equals 𝑥 squared, if we reflect this graph in the 𝑥-axis, it looks like this. And when we compare this to the remaining four graphs, we see that matches graph (A). So graph (A) must represent the equation 𝑦 equals negative 𝑥 squared.
And in fact we could’ve very quickly written off graphs (C), (D), and (E). We can see they’re not parabolas at all. In fact, they have an asymptote given by the 𝑦-axis or the line 𝑥 equals zero. That’s an indication that they’re probably some sort of reciprocal graph, for instance, 𝑦 equals one over 𝑥 squared or similar.
Now, with that in mind, let’s check our answer by using the table of values as we suggested at the beginning. This is a pretty foolproof method for plotting the graph of a quadratic function. We choose some values to substitute into our equation or function. We’re going to choose 𝑥 equals negative two, negative one, zero, one, and two. Then, we’re going to substitute 𝑥 equals negative two into the equation 𝑦 equals negative 𝑥 squared. Now, of course, this is the same as multiplying the expression 𝑥 squared by negative one. So, according to our order of operations, we’re going to square the value of 𝑥 before multiplying it by negative one. In this case, we’re going to square negative two. That gives us four. And then we’re going to multiply it by negative one, which gives us negative four. So, when 𝑥 is negative two, 𝑦 is negative four.
Let’s do the same when 𝑥 is equal to negative one. 𝑦 is negative negative one squared. So we square negative one to get one and multiply it by negative one again to get negative one. Let’s repeat this with our remaining values. When 𝑥 is zero, 𝑦 is zero. When 𝑥 is one, 𝑦 is negative one. And finally, when 𝑥 is two, 𝑦 is negative four. Plotting those on graph (A), we can see that this graph does pass through all of these five points. That confirms that the graph that represents the equation 𝑦 equals negative 𝑥 squared is (A).
