Paul's Online Notes
Paul's Online Notes
Home / Calculus I / Review / Common Graphs
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 1.10 : Common Graphs

8. Without using a graphing calculator sketch the graph of \(f\left( y \right) = {\left( {y - 1} \right)^2} + 2\).

Hint : The Algebraic transformations can also be used to help us sketch graphs of functions in the form \(x = f\left( y \right)\), but we do need to remember that we’re now working with functions in which the variables have been interchanged.
Show Solution

Even though our function is in the form \(x = f\left( y \right)\) we can still use the Algebraic transformations to help us sketch this graph. We do need to be careful however and remember that we’re working with interchanged variables and so the transformations will also switch.

In this case if we know the graph of \(h\left( y \right)\) then the graph of \(h\left( {y + c} \right) + k\) is simply the graph of \(h\left( x \right)\) shifted up by \(c\) units if \(c < 0\) or shifted down by \(c\) units if \(c > 0\) and shifted right by \(k\) units if \(k > 0\) or shifted left by \(k\) units if \(k < 0\).

So, in our case if \(h\left( y \right) = {y^2}\) we can see that,

\[f\left( y \right) = {\left( {y - 1} \right)^2} + 2 = h\left( {y - 1} \right) + 2\]

and so the graph we’re being asked to sketch is the graph of \(h\left( y \right) = {y^2}\) shifted up by 1 units and right by 2 units.

Here is the graph of \(f\left( y \right) = {\left( {y - 1} \right)^2} + 2\) and note that to help see the transformation we have also sketched in the graph of \(h\left( y \right) = {y^2}\).