Video Transcript
Choose the graph that represents the function 𝑓 of 𝑥 equals a half 𝑥 squared plus two.
There are also two further parts to this question that we’ll look at once we’ve completed the first. The function whose graph we’ve been asked to identify is a quadratic function. More specifically, it’s a function of the form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐 for constants 𝑐 and 𝑘. And such graphs have some key properties. Their shape is a symmetric parabola, with the sign of 𝑘 determining whether the parabola opens upwards or downwards. The line of symmetry of these parabolas is the 𝑦-axis. And their 𝑦-intercept, which is also the 𝑦-coordinate of the vertex of the function, is the value 𝑐.
Looking at the function 𝑓 of 𝑥 then, we can deduce that this is a symmetric parabola with the 𝑦-axis as its line of symmetry. As the coefficient of 𝑥 squared, that’s the value of 𝑘, is one-half, which is positive, we will have a parabola which opens upwards. And as the value of 𝑐 is two, this is the 𝑦-intercept of the graph. From the five options given, we can rule out graphs (C) and (E) immediately because these are each parabolas that open downwards corresponding to quadratic functions with negative values of 𝑘.
For the remaining graphs, they are all positive parabolas that open upwards and they all have the 𝑦-axis as their line of symmetry. To determine the correct graph, we need to consider the 𝑦-intercept. In graph (A), we can see that this intercepts the 𝑦-axis at the origin. So, the 𝑦-intercept is zero. Graph (B) does intercept the 𝑦-axis at a value of two, whereas graph (D) intercepts the axis somewhere between zero and one, most likely at a value of one-half. So, graph (B) represents the function 𝑓 of 𝑥 equals one-half 𝑥 squared plus two.
We’ll return to this graph in a moment, but for now let’s clear some space to answer the second part of the question. We’re now given a new set of five graphs and asked to choose the graph that represents the function 𝑓 of 𝑥 equals one-third 𝑥 squared plus four-thirds. As in the first part of the question, we once again have a quadratic function of the form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐. The value of 𝑘, that’s one-third, is positive. So, we know that this graph will be a parabola that opens upwards with the 𝑦-axis as its line of symmetry. The 𝑦-intercept of this parabola is the value of 𝑐, which is four-thirds.
We can use these properties to immediately rule out graphs (A) and (B) because these each correspond to parabolas where the value of 𝑘 is negative. Graphs (C), (D), and (E) are all positive symmetric parabolas with the 𝑦-axis as their line of symmetry. In graph (C), the 𝑦-intercept is the origin, so graph (C) is not the correct graph. In graph (D), the 𝑦-intercept is three, so graph (D) isn’t the correct graph either. In graph (E), we can see that the 𝑦-intercept is between one and two. And if we zoom in close on this graph, we can see that it is indeed at a value of four-thirds. So, the graph that represents the function 𝑓 of 𝑥 equals one-third 𝑥 squared plus four-thirds is graph (E).
In the final part of the question, we are asked, what do we find from the graphs? (a) The two curves are identical. (b) One curve is obtained by rotating the other 90 degrees about the origin. (c) The two curves appear to be the same; they are only shifted horizontally. (d) The two curves appear to be the same; they are only shifted vertically. Or (e) the second curve is a stretched form of the first curve.
So, we are now being asked to compare the two graphs, which remember represented the functions 𝑓 of 𝑥 equals a half 𝑥 squared plus two and 𝑓 of 𝑥 equals one-third 𝑥 squared plus four-thirds. I’m now going to call that second function 𝑔 of 𝑥 so we have different letters with which to refer to them. Let’s work through the five options we’ve been given. Well, firstly, the two curves clearly aren’t identical, and we can see this because at the very least they have different 𝑦-intercepts. So, it’s not option (a).
Option (b) suggests that one curve is obtained by rotating the other 90 degrees about the origin. Well, depending on the direction of rotation, if we were to rotate either graph 90 degrees about the origin, we’d obtain a graph in a different orientation. But we can see that the two curves are clearly in the same orientation, so one isn’t a rotation of the other. It’s not option (b).
Option (c) says the two curves appear to be the same, but they are only shifted horizontally. Well, one thing we can observe is that both curves have the 𝑦-axis as a line of symmetry. And if they had been shifted horizontally, they wouldn’t share the same line of symmetry. So, it’s not option (c). Option (d) says the two curves appear to be the same. They are only shifted vertically. We can determine whether this is the case by choosing some points on the two curves and considering whether the vertical difference between them is constant. For example, on the graph of 𝑓 of 𝑥, we have the point zero, two, and on the graph of the function we’re now calling 𝑔 of 𝑥, we have the point zero, four-thirds.
For the same 𝑥-value, the 𝑦-value has decreased by two-thirds. If we consider when 𝑥 is equal to negative four, we have the point negative four, 10 on the graph of 𝑓 of 𝑥. On the graph of 𝑔 of 𝑥, we can’t easily determine the 𝑦-coordinate exactly when 𝑥 is negative four, but we can see that it’s somewhere between six and seven. For the same 𝑥-value of negative four, we have to subtract at least three from the 𝑦-coordinate to go from the graph of 𝑓 of 𝑥 to the graph of 𝑔 of 𝑥. As these vertical changes aren’t constant, this means that the two graphs can’t be vertical shifts of one another.
We’re left then with only option (e), which states that the second curve is a stretched form of the first curve. We can confirm this if we consider the relationship between the functions 𝑓 of 𝑥 and 𝑔 of 𝑥 or more specifically the relationship between the coefficients. To get from the coefficient of 𝑥 squared in 𝑓 of 𝑥, which is one-half, to the coefficient of 𝑥 squared in 𝑔 of 𝑥, which is one-third, we have to multiply by two-thirds. The same is true to get from the constant term in 𝑓 of 𝑥 to the constant term in 𝑔 of 𝑥. Two multiplied by two-thirds is four-thirds. So, this tells us that two-thirds of 𝑓 of 𝑥 is equal to 𝑔 of 𝑥.
When we multiply an entire function by a constant, as we’re doing here, this corresponds to a vertical stretch of the function by that scale factor. So, 𝑔 of 𝑥, the second function, is a vertical stretch of the first by a scale factor of two-thirds. So, it’s option (e) that describes the relationship between the two graphs. The second curve is a stretched form of the first curve.
We have therefore completed the problem. We’ve identified the graphs that represent the functions 𝑓 of 𝑥 equals a half 𝑥 squared plus two and 𝑓 of 𝑥 equals a third 𝑥 squared plus four-thirds. And we’ve determined that the second curve is a stretched form of the first curve.
