Video Transcript
The following graph represents a function 𝑔 of 𝑥 after a reflection on the 𝑥-axis and a horizontal dilation by a scale factor of a half. Which of the following represents the original function 𝑓 of 𝑥? (a) 𝑓 of 𝑥 equals negative two 𝑥 minus one. (b) 𝑓 of 𝑥 equals two 𝑥 plus one. (c) 𝑓 of 𝑥 equals four 𝑥 plus two. (d) 𝑓 of 𝑥 equals negative four 𝑥 plus two. Or (e) 𝑓 of 𝑥 equals negative four 𝑥 plus one.
We’re told that this graph represents a function 𝑔 of 𝑥 after two transformations: a reflection on the 𝑥-axis and a horizontal dilation by a scale factor of a half. To find the original function 𝑓 of 𝑥, we need to reverse these transformations, starting from the one that was performed last. To reverse a horizontal dilation or stretch with a scale factor of a half, we perform a horizontal dilation with a scale factor of two. And to reverse a reflection on the 𝑥-axis, we need to reflect in the 𝑥-axis again.
So we now know the two transformations that we need to apply to 𝑔 of 𝑥 in order to find 𝑓 of 𝑥. And we can do this either graphically or algebraically. Let’s use a graphical approach first. The first transformation we’re going to apply is the horizontal dilation by a scale factor two. For a given 𝑦-coordinate, we double the corresponding 𝑥-coordinate. So each point moves twice as far horizontally from the 𝑦-axis. The pink line shows the graph of 𝑔 of 𝑥 after a horizontal dilation by a scale factor two. Next, we need to perform a reflection in the 𝑥-axis. So points that were above the 𝑥-axis now move the same distance below it, and vice versa.
The orange line now shows the function after both transformations have been reversed. So the orange line represents the function 𝑓 of 𝑥. We can now use our knowledge of the equations of straight-line graphs to find the equation of the function 𝑓 of 𝑥. We’ll use the slope–intercept form of the equation of a straight line: 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 represents the slope and 𝑏 represents the 𝑦-intercept.
From the graph, we can identify that the line crosses the 𝑦-axis at one. So the value of 𝑏 is one. We can calculate the slope of the line using the formula 𝑚 equals 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one by identifying the coordinates of two points that lie on the line. Let’s use the points zero, one and two, five. Substituting these values into the slope formula gives five minus one over two minus zero. That’s four over two, which is equal to two. So the equation of the orange line is 𝑦 equals two 𝑥 plus one. And so the function 𝑓 of 𝑥 is 𝑓 of 𝑥 equals two 𝑥 plus one. From the five options we were given, that’s option (B).
So we’ve answered the question using a graphical approach. We reversed each of the transformations graphically and then found an algebraic expression for the equation of the resulting straight line. Let’s also look at an algebraic approach. This time, we’ll find the equation of the straight line 𝑦 equals 𝑔 of 𝑥 first and then reverse the two transformations algebraically.
So let’s use our knowledge of straight lines again to find the equation of the line 𝑦 equals 𝑔 of 𝑥. From the graph, we identify that the 𝑦-intercept of this line is negative one. So the line has an equation of the form 𝑦 equals 𝑚𝑥 minus one. Using the two points with coordinates negative two, seven and zero, negative one, we can calculate the slope of this line to be seven minus negative one over negative two minus zero, which is negative four. So the equation of this line is 𝑦 equals negative four 𝑥 minus one. Or in other words, the function 𝑔 of 𝑥 is equal to negative four 𝑥 minus one.
We now need to recall the algebraic transformation that corresponds to the two transformations listed. A horizontal dilation obviously has a horizontal effect. And so it is the variable that we’re changing. 𝑥 is replaced with a half 𝑥, whereas reflection on the 𝑥-axis has a vertical effect and causes negation of the entire function. ℎ of 𝑥 is mapped to negative ℎ of 𝑥. So what we need to do is take the equation of the function 𝑔 of 𝑥 and perform each of these transformations algebraically.
Performing the horizontal dilation first, we obtain negative four multiplied by a half 𝑥 minus one, which is negative two 𝑥 minus one. Then, to reflect in the 𝑥-axis, we negate the entire function, giving negative one multiplied by negative two 𝑥 minus one, which simplifies to two 𝑥 plus one. Once again, we found that the equation of the original function 𝑓 of 𝑥 is 𝑓 of 𝑥 is equal to two 𝑥 plus one.
