Video Transcript
Given the function 𝑓 of 𝑥 equals the absolute value of 𝑥 plus three plus five, which of the following graphs is the reflection of 𝑓 of 𝑥 on the 𝑦-axis? And then we have five possible options. (A) is 𝑓 of 𝑥 equals the absolute value of 𝑥 plus three plus five. (B) 𝑓 of 𝑥 is negative the absolute value of 𝑥 plus three plus five. (C) sounds similar. 𝑓 of 𝑥 is negative the absolute value of 𝑥 plus three, but then we add five. (D) is 𝑓 of 𝑥 is the absolute value of negative 𝑥 plus three plus five. And (E) 𝑓 of 𝑥 is the absolute value of 𝑥 plus three minus five.
Now there are in fact two ways to answer this question. We could graph the function 𝑓 of 𝑥 is equal to the absolute value of 𝑥 plus three plus five and then reflect it over the 𝑦-axis. Alternatively, we can use the algebraic representation and identify what happens when we reflect a graph of 𝑓 of 𝑥 over the 𝑦-axis. We’re going to use the latter method and then use the first method to check our result.
Suppose we have the graph of the function 𝑦 equals 𝑓 of 𝑥. This graph is mapped onto the function 𝑦 equals 𝑓 of negative 𝑥 by a reflection across the 𝑦-axis. And so, given the function 𝑓 of 𝑥 equals the absolute value of 𝑥 plus three plus five, we can find the equation of the reflection of this graph across the 𝑦-axis by evaluating 𝑓 of negative 𝑥. Well, we see that every time we see an 𝑥 in the original function, we’re going to need to replace it with negative 𝑥. And so, 𝑓 of negative 𝑥 is the absolute value of negative 𝑥 plus three plus five. And in fact, we see that is option (D). So option (D) is a reflection of our function on the 𝑦-axis.
But we did say there was another method. And that was to sketch the graph of 𝑓 of 𝑥 is equal to the absolute value of 𝑥 plus three plus five and then perform the reflection. So how are we going to sketch this graph? Let’s begin by looking at the graph of the equation 𝑦 is equal to the absolute value of 𝑥 plus three. This is the graph of 𝑦 equals 𝑥 plus three, but with any values that lie below the 𝑥-axis reflected in it as shown. Then, the graph of 𝑦 is equal to the absolute value of 𝑥 plus three plus five is found by translating this graph five units up.
And we now see that if we reflect this graph across the 𝑦-axis, we end up with the red plot as shown. And this confirms that the answer is option (D). The reflection of 𝑓 of 𝑥 on the 𝑦-axis is the absolute value of negative 𝑥 plus three plus five.
