Question Video: Finding the Number of Units That a Function Is Translated in the 𝑥-Direction to Another One Mathematics

Video Transcript

𝑓 of 𝑥 is equal to three 𝑥 plus nine. This function is translated positive two units in the 𝑦-direction and 𝑐 units in the 𝑥-direction to form function 𝑔 of 𝑥, which is equal to three 𝑥 plus two. Find the value of 𝑐.

To answer this question, we first note that we are told that we translate the function 𝑓 of 𝑥 in the 𝑦-direction and 𝑥-direction to form 𝑔 of 𝑥. We can recall that we can perform these in either order. So, we can start with the 𝑦-direction. We recall that we can translate a function 𝑓 of 𝑥 𝑎 units in the 𝑦-direction by mapping 𝑓 of 𝑥 onto 𝑓 of 𝑥 plus 𝑎, where it is worth noting that the sign of 𝑎 tells us the direction of the translation. In this case, we want to translate the function two units upwards, so we set 𝑎 equal to positive two to get 𝑓 of 𝑥 plus two. We can think of this as adding two to the outputs of the function or adding two to the 𝑦-coordinates of its graph.

Similarly, we can recall that we can translate the function positive 𝑏 units in the 𝑥-direction by mapping 𝑓 of 𝑥 onto 𝑓 of 𝑥 minus 𝑏. We set 𝑏 equal to 𝑐 to get the following. We can then combine these two transformations, where it is once again worth reiterating that the order of these translations does not matter. We see that we are mapping 𝑓 of 𝑥 onto 𝑓 of 𝑥 minus 𝑐 plus two. We are told in the question that this maps onto the function 𝑔 of 𝑥, which is three 𝑥 plus two. Therefore, we must have that 𝑔 of 𝑥 is equal to 𝑓 of 𝑥 minus 𝑐 plus two.

We can then find the value of 𝑐 using this equation. First, we substitute 𝑔 of 𝑥 is equal to three 𝑥 plus two. Second, we want to find an expression for 𝑓 of 𝑥 minus 𝑐. We do this by evaluating the given function 𝑓 of 𝑥 at this expression. We do this by replacing 𝑥 with 𝑥 minus 𝑐 in the given expression to get three times 𝑥 minus 𝑐 plus nine. Third, we add on our constant of two. So we have three 𝑥 plus two is equal to three times 𝑥 minus 𝑐 plus nine plus two.

We can now solve this equation for 𝑐. We start by adding nine to two to get three 𝑥 plus two equals three times 𝑥 minus 𝑐 plus 11. We then subtract 11 from both sides of the equation. This gives us three 𝑥 minus nine is equal to three times 𝑥 minus 𝑐. We can then distribute the factor of three over the parentheses. This yields three 𝑥 minus nine is equal to three 𝑥 minus three 𝑐. Then, we subtract three 𝑥 from both sides of the equation. This gives us negative nine is equal to negative three 𝑐. Finally, we divide the equation through by negative three. This then gives us that the value of 𝑐 must be equal to three.