Question Video: Finding the Resulting Functions after Two Transformations Have Been Applied Mathematics

Video Transcript

The following graph represents a function 𝑓 of 𝑥. Find the resulting function 𝑔 of 𝑥 after a reflection on the 𝑦-axis followed by a translation by two units in the positive direction of the 𝑥-axis.

To find the function 𝑔 of 𝑥, we need to apply two transformations to the function 𝑓 of 𝑥: first, a reflection on the 𝑦-axis and, second, a translation two units in the positive direction of the 𝑥-axis. We can do this either graphically or algebraically and we’ll consider both approaches. Let’s use the algebraic approach first. We’re going to determine the equation of the function 𝑓 of 𝑥 and then apply the two transformations algebraically. To find the equation of the graph of 𝑓 of 𝑥, we observe that it is a parabola, with its vertex at the point with coordinates negative three, negative four. This tells us that the equation of the function 𝑓 of 𝑥 must be of the form 𝑓 of 𝑥 equals 𝑘 multiplied by 𝑥 plus three squared minus four for some constant 𝑘.

To determine the value of 𝑘, we can use the coordinates of any other point on the curve. Let’s use the point negative one, zero. This tells us that when 𝑥 is equal to negative one, 𝑓 of 𝑥 is equal to zero. So substituting negative one for 𝑥 and zero for 𝑓 of 𝑥, we have zero is equal to 𝑘 multiplied by negative one plus three squared minus four. Negative one plus three is two and two squared is four, so we have zero is equal to four 𝑘 minus four. We can then add four to each side of the equation and divide both sides by four to find that 𝑘 is equal to one. So, the equation of the function 𝑓 of 𝑥 is 𝑓 of 𝑥 is equal to 𝑥 plus three squared minus four.

Next, we need to consider the effect of applying each of these transformations on the equation of the function. Reflection in the 𝑦-axis first of all has a horizontal effect, and it corresponds to a change of the variable. 𝑥 is replaced with negative 𝑥. So, following reflection on the 𝑦-axis, the function which at this point we’ll refer to as ℎ of 𝑥 is equal to negative 𝑥 plus three squared minus four. A translation by two units in the positive direction of the 𝑥-axis also has a horizontal effect, and it corresponds to a change of the variable, this time from 𝑥 to 𝑥 minus two. So, replacing 𝑥 with 𝑥 minus two, the function which is now 𝑔 of 𝑥 is equal to negative 𝑥 minus two plus three all squared minus four. Distributing the inner parentheses gives negative 𝑥 plus two plus three squared minus four. And then simplifying, we have negative 𝑥 plus five squared minus four.

So, by first finding the equation of the graph of 𝑓 of 𝑥 and then applying the two transformations algebraically, we found that the function 𝑔 of 𝑥 is given by negative 𝑥 plus five squared minus four. Let’s now consider a graphical approach. First, we need to reflect the graph of 𝑦 equals 𝑓 of 𝑥 in the 𝑦-axis. We can consider some key points. Firstly, the vertex of the graph, which was at negative three, negative four, will now be at positive three, negative four. The 𝑥-intercepts of the graph, which were negative one and negative five, will now be positive one and positive five. And the 𝑦-intercept of five will be unchanged. So the graph in orange represents the function after a reflection in the 𝑦-axis.

Next, we need to transform this function by applying a translation of two units in the positive direction of the 𝑥-axis. Every point moves two units to the right. So, we now have the graph of the function 𝑦 equals 𝑔 of 𝑥 shown in pink. We can find the equation of this graph in the same way as we did for the graph of 𝑓 of 𝑥. The vertex is at the point five, negative four. So, the equation of 𝑔 of 𝑥 is of the form 𝑘 multiplied by 𝑥 minus five squared minus four.

We can then use the coordinates of any other point on the curve to determine the value of 𝑘. Using the point with coordinates three, zero, we obtain zero is equal to 𝑘 multiplied by three minus five squared minus four. This leads to zero equals four 𝑘 minus four, and solving as before, we find that 𝑘 is equal to one. So, the equation of 𝑔 of 𝑥 is 𝑔 of 𝑥 equals 𝑥 minus five squared minus four.

Now, this doesn’t look exactly like the equation for 𝑔 of 𝑥 that we found using our previous method. However, we can make the two identical by factoring by negative one inside the parentheses. 𝑥 minus five is equivalent to negative negative 𝑥 plus five. Then, because we’re squaring, we can write this as negative one squared multiplied by negative 𝑥 plus five squared minus four, but of course negative one squared is simply one. So, we’ve rewritten 𝑔 of 𝑥 as negative 𝑥 plus five squared minus four, which agrees with our previous answer.

Using two methods then, applying the transformations algebraically and then applying the transformations graphically, we found that the function 𝑔 of 𝑥, which is obtained from the function 𝑓 of 𝑥 by a reflection on the 𝑦-axis and then a translation two units in the positive direction of the 𝑥-axis, is 𝑔 of 𝑥 is equal to negative 𝑥 plus five squared minus four.