Question Video: Identifying the Graph of a Quadratic Function and Determining the Relationship between Two Quadratic Graphs Mathematics

Video Transcript

Answer the following questions. Which graph represents the quadratic function 𝑓 of 𝑥 equals 𝑥 squared plus three? Which graph represents the quadratic function 𝑓 of 𝑥 equals 𝑥 squared plus four?

There is also a third part to this question, which we will look at when we’ve completed the first two parts. To answer the first part of the question then, let’s think about the properties of the graph of the function 𝑓 of 𝑥 equals 𝑥 squared plus three. First, this is a quadratic function. So we know that the shape of its graph will be a parabola. The parabola will open upwards if the coefficient of 𝑥 squared is positive, and it will open downwards if the coefficient of 𝑥 squared is negative. In this question, the coefficient of 𝑥 squared is one, which is positive. So we know that the shape of the graph will be a parabola that opens upwards, or we might say a U-shaped parabola.

Now, all parabolas are symmetric with a vertical line of symmetry. But furthermore, we know that any quadratic function of the form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐, which is what we have here, is a parabola with its line of symmetry on the 𝑦-axis.

Next, let’s consider the 𝑦-intercept of the graph of this function. We know that the 𝑦-intercept occurs when 𝑥 is equal to zero because 𝑥 is equal to zero everywhere on the 𝑦-axis. We can therefore find the 𝑦-value of the 𝑦-intercept by substituting 𝑥 equals zero into the function, or in other words evaluating 𝑓 of zero. We have zero squared plus three, which is equal to three. We know then that the coordinates of the 𝑦-intercept for the given quadratic function are zero, three.

Now, in fact, as the line of symmetry is the 𝑦-axis and the line of symmetry passes through the vertex of a quadratic function, we know that the vertex will also have the coordinates zero, three. We now look at the five graphs we were given. Graph (B) is a U-shaped parabola with its line of symmetry along the 𝑦-axis and the 𝑦-intercept and vertex with coordinates zero, three. So graph (B) is the correct graph.

If we were to look at the others, we could rule graph (E) out because it is a negative parabola, that is, a parabola that opens downwards. And we could rule out options (A), (C), and (D) because they all have 𝑦-intercepts that are not the point zero, three. So we’ve answered the first part of the question and found the graph which represents the quadratic function 𝑓 of 𝑥 equals 𝑥 squared plus three.

Let’s now look at the second part, in which we’re finding the graph which represents the quadratic function 𝑓 of 𝑥 equals 𝑥 squared plus four. Well, for the same reasons as in the first part of the question, we know that this will be a positive parabola, or a parabola which opens upwards, with its line of symmetry along the 𝑦-axis. To find the 𝑦-intercept and the coordinates of the vertex, we evaluate 𝑓 of zero, giving zero squared plus four, which is four. So the coordinates of the 𝑦-intercept and the coordinates of the vertex are zero, four.

We now know that we’re looking for a parabola that opens upwards with a line of symmetry along the 𝑦-axis and a 𝑦-intercept and vertex with coordinates of zero, four. Looking at the five graphs, we can see that the one which has these correct properties is graph (D).

We’ll now clear some space to answer the final part of the question.

Which of the following is true about the two graphs? (A) The two curves are identical. (B) The first curve is just a stretched form of the second curve. (C) The two curves have the same shape, but the second is a horizontal shift of the first. (D) The two curves have the same shape, but the second is a vertical shift of the first. Or (E) one curve is obtained by rotating the other by 90 degrees about the origin.

We have here the two graphs that we identified earlier in the question: the graph representing the function 𝑓 of 𝑥 equals 𝑥 squared plus three and the graph representing the function 𝑓 of 𝑥 equals 𝑥 squared plus four. We can see by looking at the two graphs side by side that they do have the same shape. But the graph of 𝑓 of 𝑥 equals 𝑥 squared plus four is above the graph of 𝑓 of 𝑥 equals 𝑥 squared plus three. If we label the vertex of each graph, we have zero, three for the first function and zero, four for the second. So the vertex of the graph of the function 𝑓 of 𝑥 equals 𝑥 squared plus four is one unit vertically above the vertex of the graph of the first function.

We can also see this if we label other points on the two curves. The point two, seven is on the graph of 𝑓 of 𝑥 equals 𝑥 squared plus three. And the point two, eight is on the graph of 𝑓 of 𝑥 equals 𝑥 squared plus four. So, for the same 𝑥-coordinate, the 𝑦-coordinate, or the value of the function, is one more for the second function than for the first. If we write the second function as 𝑥 squared plus three plus one, then this confirms that we are indeed just adding one to the first function. And from our knowledge of transformations of graphs, we know that adding a constant to a function is a vertical shift or vertical translation by that constant. So the relationship between the two graphs is that the two curves have the same shape, but the second is a vertical shift of the first. And in fact it is a vertical shift by one unit.

Looking quickly at the other options, we can see that none of them are also true. Firstly, option (a), the two curves are not identical; they don’t have the same vertex or indeed any of the same points. The two curves have the exact same shape. So the first curve is not a stretched form of the second. Considering option (c), the two curves do have the same shape as we’ve already said. But as the vertex of each graph has the same 𝑥-coordinate, we can see that the second is not a horizontal shift of the first. And finally, in (e), if one curve was obtained by rotating the other by 90 degrees about the origin, then the second curve would be a parabola in a different orientation. And as we can see, the two parabolas are in exactly the same orientation.

So we have identified the graph that represents the function 𝑓 of 𝑥 equals 𝑥 squared plus three and the graph that represents the function 𝑓 of 𝑥 equals 𝑥 squared plus four. And we’ve found that the relationship between the two graphs is that the two curves have the same shape, but the second is a vertical shift of the first.