Which of the following graphs
represents the equation 𝑓 of 𝑥 equals 𝑥 plus four times 𝑥 minus two? Options A, B, C, D, and E.
In this question, we are given a
function 𝑓 of 𝑥 and asked to determine which of the five given graphs represent
this function. We can note that 𝑓 of 𝑥 is the
product of two linear factors, and if we were to expand this expression, we would
obtain a quadratic.
We could answer this question by
eliminating options; however, it is a more useful skill to be able to sketch
quadratics from their equations. To do this, we first note that for
a product to be equal to zero, one of the factors must be equal to zero. This means that the function 𝑓 of
𝑥 can only be equal to zero when 𝑥 equals negative four or 𝑥 equals two. If the function outputs zero at
these values of 𝑥, then they must be the 𝑥-intercepts of its graph. This is a good starting point to
begin our sketch, but we can determine more information about the graph of this
function by expanding the product.
We can do this by finding the sum
of the product of each pair of terms from the two factors. We find that 𝑓 of 𝑥 is equal to
𝑥 squared plus four 𝑥 minus two 𝑥 minus eight. This gives us 𝑥 squared plus two
𝑥 minus eight. We now see that 𝑓 of 𝑥 is a
quadratic equation with leading coefficient one and constant term negative
We can then recall that if a
quadratic function has a positive leading coefficient, then its graph will be a
parabola that opens upwards. In other words, it is U-shaped. We can also recall that the
constant term in the function tells us the value of the 𝑦-intercept, since we can
substitute 𝑥 equals zero into the function to be left with only the constant
This is now enough information to
sketch the parabola. First, we can add the 𝑥-intercepts
at negative four and two. Second, we can add the 𝑦-intercept
at negative eight onto our sketch. Finally, we connect these points
with a parabolic shape that opens upwards to get the following sketch. We can now compare this sketch to
each of the five given options.
In option (A), we can note there
are no 𝑥-intercepts and the 𝑦-intercept is not at negative eight, so this is not
the correct graph. In option (B), we can see from the
graph that the 𝑦-intercept is not at negative eight and that the parabola opens
downwards, not upwards, so this cannot be the correct graph. In option (C), we see that the
𝑥-intercepts are not at negative four and two, and the 𝑦-intercept is not at
negative eight. So, this is not the correct
In option (D), we can see that the
𝑥-intercepts are not at negative four and two. So, this cannot be the correct
graph. Finally, in option (E), we can see
that 𝑥- and 𝑦-intercepts are in the correct places and we also have a parabolic
curve which opens upwards, so this must be the correct option.
Hence, the graph of 𝑓 of 𝑥 equals
𝑥 plus four times 𝑥 minus two is given by option (E).