Question Video: Identifying the Graph of a Quadratic Function Mathematics • Third Year of Secondary School

Video Transcript

Which of the following is the graph of 𝑓 of 𝑥 is equal to two 𝑥 squared plus five? Options (A), (B), (C), (D), and (E).

In this question, we’re given five graphs. And we need to determine which of the five graphs represents the function 𝑓 of 𝑥 is equal to two 𝑥 squared plus five. And there’s many different ways we could go about answering this question. For example, we could analyze the function 𝑓 of 𝑥 to determine how to sketch the graph 𝑦 is equal to 𝑓 of 𝑥. And there’s a lot of different things we could check about this function. For example, we could analyze its first and second derivatives to determine its intervals of increase and decrease, its critical points, and its convexity. And while this would work, it’s very complicated. And we can see we’re only dealing with a quadratic.

And since we have no term of degree one in our quadratic function, we can sketch the graph of this function entirely in terms of transformations. We can do this by calling the function 𝑥 squared 𝑔 of 𝑥. We can then note that 𝑓 of 𝑥 is equal to two times 𝑔 of 𝑥 plus five. We multiply the outputs of 𝑔 of 𝑥 by two and then we add five to these outputs. Multiplying the outputs of 𝑔 of 𝑥 by two will be a vertical stretch by a factor of two and then adding five to the outputs of these functions will be a vertical translation of five units in the positive direction. Therefore, we can sketch 𝑦 is equal to 𝑓 of 𝑥 by applying these transformations to the graph 𝑦 is equal to 𝑥 squared.

We know that this curve has a parabolic shape opening upwards and it passes through the origin. Its sketch looks like the following. Multiplying the outputs of the function by two will stretch its graph vertically by a factor of two. Every point on the curve doubles its displacement from the 𝑥-axis; we get the following. We then want to add five onto the outputs of this function. This will translate its graph five units upwards. This gives us the following sketch. And we can find the coordinates of the 𝑦-intercept of this curve.

Stretching the curve vertically will not affect its 𝑦-intercept and then translating it five units upwards will translate the 𝑦-intercept five units up. The 𝑦-intercept of 𝑦 is equal to 𝑥 squared was at zero, zero. So translating this point five units up, the 𝑦-intercept will be the point zero, five. Therefore, our sketch must be a parabolic shape. So it can’t be option (C) or (D).

We know the parabola must open upwards, so it can’t be option (A). And we know its 𝑦-intercept is at zero, five, so it can’t be option (B). This just leaves option (E), which matches our sketch. Therefore, we were able to show, of the following five given graphs, only option (E) represents the graph of the function 𝑓 of 𝑥 is equal to two 𝑥 squared plus five.