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I've been teaching high school math for over 20 years. I write about my experiences here and also share activities that I have done with my classes that have been successful (and some that have been not so successful).
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How to Teach Graphing Transformations of Functions

When teaching transformations, it can sometimes be confusing for students (and teachers) when there are multiple transformations on a function and students arrive at a different result based on the order in which they did the transformations.

In this blog post, I'm going to show you how I teach graphing transformations to Precalculus students.

First, students can do some exploration in Desmos or a graphing calculator to determine how different parameters in the equation of a function affect the graph of the function.

Guide the students to summarize their findings in a chart such as this:

Students should notice that some transformations affect the graph vertically and some affect the graph horizontally.

 Transformations that affect the graph VERTICALLY Transformations that affect the graph HORIZONTALLY vertical shift reflection in the x-axis vertical dilation (stretch or shrink) horizontal shiftreflection in the y-axishorizontal dilation (stretch or shrink)

If a function has a transformation that affects the graph horizontally in more than one way, then THE ORDER OF THE TRANSFORMATION MATTERS.  The same is true for more than one transformation that affects the graph vertically.

You may also be interested in my blog post: How to Teach Graphing Sine and Cosine

One vertical, one horizontal

First, let's look at some easy examples with only two transformations, one affecting the graph vertically, and one affecting the graph horizontally.

 $y=\sqrt{x-2}+1$ horizontal shift to the right 2, vertical shift up 1 $y=\sqrt{2x}+1$ horizontal stretch by a factor of 2, vertical shift up 1 $y=\sqrt{-x}+1$ reflection in the $y$-axis, vertical shift up 1 $y=-\sqrt{x-2}$ reflection in the $x$-axis, horizontal shift right 2 $y=3\sqrt{x-2}$ vertical stretch by a factor of 3, horizontal shift right 2

For the above examples, it does not matter the order in which you do the transformation. For example, with the last function, you could shift to the right first, and then do the vertical stretch or vice versa. Either way, you will end with the same result.

More than one vertical, more than one horizontal

However, if a function has a transformation that affects the graph horizontally (or vertically) in more than one way, then THE ORDER OF THE TRANSFORMATION MATTERS.

Encourage students to write the function in the form $y=af(b(x-c))+d$ to help them determine the correct transformation. In other words, students should change   $y=|-x+3|$   to   $y=|-(x-3)|$

Do the sequence of transformations in this order: Dilate, Reflect, Shift (DRS).

Dilate (stretch or shrink; the vertical or horizontal order does not matter)
Reflect (the $x$-axis or $y$-axis order does not matter)
Shift (the vertical or horizontal order ddoeso not matter)

Let's explore some functions that have transformations that affect the graph horizontally (or vertically) in more than one way.

Example 1:
$y=-2\sqrt{x}+1$

When teaching graphing transformations, I ask the students to pick a few values from the parent graph and put them in a t-table.

They then determine how each transformation will affect the $x$- or $y$-coordinates.

For this example, $y=-2\sqrt{x}+1$, students should be able to determine that...

(DILATION) the vertical stretch by 2 will affect the $y$-coordinates, so they should multiply the $y$-values from the parent graph by 2.

(REFLECTION) the reflection in the $x$-axis will cause the signs of all of the $y$-coordinates to change, so they should change the signs of the $y$-values.

(SHIFT) the vertical translation up 1 indicates that they should add 1 to each of the $y$-values.

Students can either do this in a series of t-tables (see image below) or do them all at once to in one final t-table.

Example 2:
$y=\sqrt{-2(x+3)}$

This function has three transformations that affect the graph horizontally.

(DILATION) the horizontal shrink by a factor of $1/2$ will affect the $x$-coordinates; students should multiply the $x$-values from the parent graph by $1/2$.

(REFLECTION) the reflection in the $y$-axis will cause the signs of all of the $x$-coordinates to change; students should change the signs of the $x$-values.

(SHIFT) the horizontal translation left 3 indicates that students should subtract 3 from each of the $x$-values.

You may have determined by looking at my written work, that you could condense some of the steps and write fewer t-tables. In example 2, you could have done the horizontal shrink and reflection in the $x$-axis at the same time.

Example 3:
$y=-2\sqrt{-2(x+3)}+1$

Here is an example that has six transformations: three that affect the graph horizontally and three that affect the graph vertically.

In my written work below, notice how I condensed my work in order to not have six t-tables. In green, I did both of the dilations, in purple I did both of the reflections, and in red I did both of the shifts.

Remember that the function must be written in the form: $y=af(b(x-c))+d$

Notice in example 3 above that I have $y=\sqrt{-2(x+3)}$ instead of $y=\sqrt{-2x-6}$

In other words, for the horizontal translation, one must look at whatever is subtracted from $x$, not whatever is subtracted from $2x$ or $-2x$ or even $-x$.

Example 4:
You can even use this sequence of transformations when given the graph of a piecewise function. Look at this piecewise function below and how I transformed it to $g(x)=-\frac{1}{2} f(2(x-1)-3$

How to graph transformations in Desmos

Below is an image showing the graph of a transformed function using Desmos. I named the parent graph $f(x)$, then in the subsequent line I am able to refer to $f(x)$

Notice that you can't see the green function because it's underneath the purple function (because they are identical).

IS THIS THE ONLY WAY?

NO. You could take the approach that you simply maintain an anchor point as you transform the function and do reflections at its anchor point.

For example, if given $y=f(-(x-2))$, you could first shift right 2 units, then reflect in the line $x=2$ instead of reflecting in the $y$-axis.

You may also be interested in my blog post: How to Teach Graphing Sine and Cosine

Do you have a different way of teaching transformations that has been successful? Join me in my Facebook group. and let me know.

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