Video Transcript
Which of the following graphs represents the equation 𝑦 equals 𝑥 squared? And then, we have five graphs labeled (A) through (E) to choose from.
Now there are in fact a couple of different ways that we can answer this question. One of these is to create a table of values. The other is to use what we know about quadratic graphs. Let’s use this method and then use a table of values to check. Recall, a quadratic graph is a symmetric parabola with a vertical line of symmetry that runs through its vertex or its turning point. The most basic quadratic equation has the equation 𝑦 equals 𝑥 squared.
Now of course the graph of this equation is a symmetric parabola, and it has a turning point or a vertex at the origin. That’s the point with coordinates zero, zero. We see then that we can instantly disregard three of our graphs. That’s the graphs (A), (B), and (C). None of these graphs are parabolas. In fact, these are the graphs of some reciprocal function, for instance, one over 𝑥 squared. Graphs (D) and (E), however, are both parabolas, and they both pass through the origin.
So which out of (D) and (E) is the graph with equation 𝑦 equals 𝑥 squared? Well, there is an extra bit of information about the graph of 𝑦 equals 𝑥 squared. And that is that its parabola opens upwards. That’s option (E). So option (E) has equation 𝑦 equals 𝑥 squared. In fact, the equation of graph (D) is 𝑦 equals negative 𝑥 squared. This is a reflection of the graph of 𝑦 equals 𝑥 squared in the 𝑥-axis, and this opens downward.
So we did mention that there was a second technique we could have used, and that is to plot a table of values. We’re going to choose the values 𝑥 equals negative two, negative one, zero, one, and two to substitute into our equation. If 𝑥 is equal to negative two, 𝑦 is equal to negative two squared, which is four. Similarly, when 𝑥 is negative one, 𝑦 is negative one squared, and that’s one. Substituting 𝑥 equals zero, one, and two into our equation, and we get 𝑦 equals zero, one, and four, respectively. Extracting each pair of values as a pair of coordinates, and we see that once again the graph that passes through each of these points is graph (E). Graph (E) represents the equation 𝑦 equals 𝑥 squared.
