Complete the following table for the graph of 𝑓 of 𝑥 equals 0.5 minus two 𝑥 squared by finding the values of 𝑎, 𝑏, 𝑐, and 𝑑. Which figure represents the graph of the function 𝑓 of 𝑥?
We’ve been given the definition of a function 𝑓 of 𝑥. 𝑓 of 𝑥 is equal to 0.5 minus two 𝑥 squared. The table gives integer values of 𝑥 from negative two to positive two in the top row. And then the values of the function 𝑓 for each value of 𝑥 are given in the second row. But the second row is incomplete. The values of 𝑓 of 𝑥 for 𝑥 equals negative two, negative one, zero, and positive one are represented by the letters 𝑎, 𝑏, 𝑐, and 𝑑, respectively, which we need to determine.
To answer this question then, we need to evaluate the function 𝑓 for each integer value of 𝑥 from negative two to one. Starting with negative two then, we have 𝑓 of negative two is equal to 0.5 minus two multiplied by negative two squared. That’s 0.5 minus two multiplied by positive four or 0.5 minus eight, which is negative 7.5. The value of 𝑎 then, found by evaluating 𝑓 of negative two, is negative 7.5.
To find the value of 𝑏, we evaluate 𝑓 of negative one, which is 0.5 minus two multiplied by negative one squared. That’s 0.5 minus two multiplied by one or 0.5 minus two, which is negative 1.5. So, we’ve also found the value of 𝑏. Next, we want to determine the value of 𝑐, which we can do by evaluating 𝑓 of zero. It’s 0.5 minus two multiplied by zero squared. Of course, zero squared is just zero, and multiplying anything by zero still gives zero. So, we have 0.5 minus zero, which is 0.5.
Finally, we determine the value of 𝑑 by evaluating 𝑓 of one, which is 0.5 minus two multiplied by one squared. That’s. 0.5 minus two multiplied by one, 0.5 minus two, which is negative 1.5.
We’ve therefore found the values of the four unknowns. 𝑎 is equal to negative 7.5, 𝑏 is equal to negative 1.5, 𝑐 is equal to 0.5, and 𝑑 is equal to negative 1.5.
Let’s now consider the second part of the question, in which we’re asked which figure represents the graph of the function 𝑓 of 𝑥. Now, we could plot each of the points we’ve just determined and connect them with a curve. Or instead, we could consider the key features of the graph of 𝑦 equals 𝑓 of 𝑥.
First, this is a quadratic curve, so its shape will be a parabola. Next, we know that the leading coefficient, that’s the coefficient of 𝑥 squared, is negative. So, we will have a parabola that opens downwards, which we might call an n-shaped parabola. Looking at the five options we’ve been given, we can see that they are indeed all parabolas. But we can rule out option (D) on the basis that it is a U-shaped parabola which opens upwards and therefore corresponds to a quadratic curve with a positive leading coefficient. All four of the other curves though do correspond to parabolas with negative leading coefficients. So, we need to keep going.
In our working for the first part of the question, we found that 𝑓 of zero is equal to 0.5. Now, in general substituting the value 𝑥 equals zero gives the 𝑦-intercept of the curve because 𝑥 is equal to zero on the 𝑦-axis. We therefore know that the graph of 𝑓 of 𝑥 intercepts the 𝑦-axis at 0.5. On the basis of this, we can rule out options (A) and (B) because they each pass through the origin, and so they intercept the 𝑦-axis at a value of zero.
For the same reason, we can rule out option (C) because this intercepts the 𝑦-axis at a negative 𝑦-value, potentially negative 0.5. Looking at the final graph, graph (E), we can see that this does indeed intercept the 𝑦-axis at a value approximately halfway between zero and one. So, it seems reasonable that this value is 0.5.
To confirm our answer, we could also check that some of the other points in the table do indeed lie on the curve. For example, the point negative one, negative 1.5 and the point positive one, negative 1.5 do both lie on this curve. We can’t see the 𝑦-coordinates when 𝑥 is equal to negative two and positive two on the graph, but we can at least deduce that they are less than negative five. And so, this is consistent with the values of negative 7.5 in the table.
So, we’ve completed the problem. We found that the values of 𝑎, 𝑏, 𝑐, and 𝑑 are 𝑎 equals negative 7.5, 𝑏 equals negative 1.5, 𝑐 equals 0.5, and 𝑑 equals negative 1.5. And the figure which represents the graph of the function 𝑓 of 𝑥 is graph (E).