Question Video: Finding the Equation of a Function When Given the Graph of Its Reflection an Axis Mathematics

The following linear graph represents a function 𝑔(𝑥) after a reflection in the 𝑦-axis. Find the original function 𝑓(𝑥).

02:31

Video Transcript

The following linear graph represents a function 𝑔 of 𝑥 after a reflection in the 𝑦-axis. Find the original function 𝑓 of 𝑥.

There are a few ways to approach this problem. One way would be to find the equation of the function 𝑔 of 𝑥 given on the graph. As it is a linear function, we know it can be written in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope and 𝑏 is the 𝑦-intercept. It is clear from the graph that the 𝑦-intercept is negative, four. We know that the slope or gradient of any line is equal to the rise over the run. Choosing the two points negative four, four and negative two, zero which lie on the line, the slope is equal to negative four over two. This is equal to negative two. The function 𝑔 of 𝑥 has equation negative two 𝑥 minus four. We are told that 𝑔 of 𝑥 is a reflection of 𝑓 of 𝑥 in the 𝑦-axis. Therefore, 𝑓 of 𝑥 is also a reflection of 𝑔 of 𝑥 in the 𝑦-axis.

This means that 𝑓 of 𝑥 is equal to 𝑔 of negative 𝑥. The function 𝑓 of 𝑥 is therefore equal to negative two multiplied by negative 𝑥 minus four. This simplifies to two 𝑥 minus four. The original linear function 𝑓 of 𝑥 has equation two 𝑥 minus four. An alternative method would have been to have sketched the reflection of 𝑔 of 𝑥 on the graph. When reflecting a graph in the 𝑦-axis, we know that the 𝑦-intercept remains the same and the roots change signs. This means that 𝑓 of 𝑥 will still intercept the 𝑦-axis at negative four and intercept the 𝑥-axis at positive two. As the slope or gradient of this line is two and the 𝑦-intercept is negative four, the equation of the line is two 𝑥 minus four. This confirms the answer we found using our first method.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.