Question Video: Graphing Linear Functions by Making Tables Mathematics

Video Transcript

By making a table of values, determine which of the following is the function represented by the graph shown. Is it (A) 𝑓 of 𝑥 is equal to negative four 𝑥? (B) 𝑓 of 𝑥 is equal to negative four 𝑥 plus one. (C) 𝑓 of 𝑥 is equal to four 𝑥. (D) 𝑓 of 𝑥 is equal to four 𝑥 minus one. Or (E) 𝑓 of 𝑥 is equal to four 𝑥 plus one.

We begin by recalling that as our graph is a straight line, it is a linear function of the form 𝑚𝑥 plus 𝑏, where 𝑚 is the slope or gradient of the function and 𝑏 is the 𝑦-intercept. We are asked to determine which of the five functions is represented by the graph by making a table of values. From the graph, we see that there are three points with integer or whole-number 𝑥- and 𝑦-coordinates. They are the points one, four; zero, zero; and negative one, negative four. Starting with the smallest 𝑥-value, we can place these in a table of values as shown. We can now use these values to identify the correct function. We will begin by substituting 𝑥 equals zero into the five functions.

In option (A), 𝑓 of zero is equal to negative four multiplied by zero. As this is equal to zero, the point zero, zero does satisfy the function 𝑓 of 𝑥 is equal to negative four 𝑥. Repeating this for option (B), we have 𝑓 of zero is equal to negative four multiplied by zero plus one. As this is equal to one, the point zero, zero does not satisfy this function, and option (B) is incorrect. In option (C), 𝑓 of zero is equal to zero. So, the point zero, zero does lie on the graph of the function 𝑓 of 𝑥 is equal to four 𝑥. In option (D), 𝑓 of zero is equal to negative one. This means that the point zero, zero does not satisfy the function. The same is true of option (E), where 𝑓 of zero is equal to one.

We now have only two possible answers, option (A) and option (C). Let’s now consider the value of 𝑓 of 𝑥 when 𝑥 is equal to one. In option (A), 𝑓 of one is equal to negative four multiplied by one, which is equal to negative four. And in option (C), 𝑓 of one is equal to four multiplied by one, which equals four. As the point one, four and not one, negative four lies on the graph, we know that option (A) is incorrect. We have proved that the point zero, zero and one, four lie on the graph of the function 𝑓 of 𝑥 is equal to four 𝑥.

Let’s now consider the third point from our table and substitute 𝑥 equals negative one into our function. Four multiplied by negative one is negative four. So the point with coordinates negative one, negative four does lie on the graph of the function. We can therefore conclude that option (C) 𝑓 of 𝑥 is equal to four 𝑥 is the correct function.

It is worth noting that we could’ve eliminated the other four options directly from the graph. As our line slopes upwards from left to right, we know that the slope or gradient is positive. This rules out options (A) and (B). Next, since the graph passes through the origin, we know that the 𝑦-intercept is equal to zero. And this rules out options (D) and (E) together with option (B) once again, as these three functions have a 𝑦-intercept of negative one, one, and one, respectively. Once again, this would leave us with the correct answer option (C). 𝑓 of 𝑥 is equal to four 𝑥.