Question Video: Completing a Table of Values for a Quadratic Function Mathematics

Complete the following table for the graph of 𝑓(𝑥) = 2𝑥² by finding the unknowns 𝑎 and 𝑏.

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Video Transcript

Complete the following table for the graph of 𝑓 of 𝑥 equals two 𝑥 squared by finding the unknowns 𝑎 and 𝑏.

We’ve been given the function 𝑓 of 𝑥 equals two 𝑥 squared and asked to complete a table of values that could then be used to plot the graph of 𝑓 of 𝑥. The completed table will contain ordered pairs of 𝑥, 𝑓 of 𝑥 values that will give the coordinates of points that lie on the graph of 𝑓 of 𝑥. For example, we’re given that when 𝑥 is equal to negative two, 𝑓 of 𝑥 is equal to eight. So the point with coordinates negative two, eight lies on the graph of 𝑓 of 𝑥, as do the points with coordinates zero, zero and one, two.

We are asked to find the unknowns 𝑎 and 𝑏, which, we can see, are the 𝑓 of 𝑥 values associated with 𝑥-values of negative one and positive two. In other words, 𝑓 of negative one is equal to 𝑎, and 𝑓 of two is equal to 𝑏. We can therefore find the values of 𝑎 and 𝑏 by substituting 𝑥 equals one and then 𝑥 equals two into the function 𝑓 of 𝑥.

To find 𝑎 first of all then, we substitute 𝑥 equals negative one. 𝑓 of negative one is two multiplied by negative one squared. That’s two multiplied by one, which is equal to two. And so we find that 𝑎 is equal to two. To find 𝑏, we substitute 𝑥 equals two into the function 𝑓 of 𝑥. 𝑓 of two is equal to two multiplied by two squared. That’s two multiplied by four, which is equal to eight. And so we’ve also found the value of 𝑏.

We now know that the points with coordinates negative one, two and two, eight also lie on the graph of 𝑓 of 𝑥. If we wanted, we could plot these five coordinates on a coordinate grid and then join the points together with a curve to give the graph of 𝑓 of 𝑥.

We’ve answered the question, though, which was to determine the values of the unknowns 𝑎 and 𝑏. 𝑎 is equal to two, and 𝑏 is equal to eight.