Monday 25 September class

Posted on September 25, 2017 by Sybil Shaver

Topics:

• More on functions defined by graphs (Example 3.12)

• Basic graphs and transformations: here is the handout (outline notes to be filled in, plus the basic graphs) MAT1375BasicsFunctionsTransformationsOutlineSummary

You can see how I filled out the Outline Summary here: but you should fill it out using your own words after playing with the Desmos graphs! MAT1375BasicsFunctionsTransformationsMySummary

Here are the Desmos graphs that I used to show the effects of the various transformations:

We considered what would happen if we change a function by adding a number c to its output, in other words, if the basic function is f(x), we make a new function g(x) = f(x) +c. My Desmos graph uses the basic quadratic function y = x^{2}, so we are considering the graphs of functions of the form y= x^{2} + c. By using desmos to play around with different values for c, we found that this moves the graph upward or downward by the amount c, depending on whether c is positive or negative. The graph with sliders to change the value of c is here (in desmos).

We next considered what would happen if we change the function by adding a number c to its input, in other words, we substitute (x+c) in place of x in the formula for the function. So now, using the same basic function y=x^{2}, we are considering the graphs of functions of the form y=(x+c)^{2}. By using desmos to play around with different values for c, we found that this moves the graph to the left or to the right by the amount c, depending on whether c is positive or negative: it moves to the left if c is positive, and to the right if c is negative. The graph with sliders to change the value of c is here (in desmos).

Next, we considered what would happen if we change the function by multiplying the output by a positive number a$. Since it can be hard to see exactly what is happening and what is different between multiplying the output by a and multiplying the input by a, we are starting with the graph of y = x^{3} +1, and we will consider the graph of y = a(x^{3}) +1 for various values of a which are positive. Notice that the original function has a y-intercept at (o,1) and also that its graph passes through the points (-1, 0) and (1, 2). Pay attention to what happens to those points when we make changes. By using desmos to play around with different values for a which were positive, we found that this transformation stretches or compresses the graph toward the x-axis, depending on whether a is greater than 1 or a is between 0 and 1 .   The graph with sliders to change the value of a is here (in desmos). [As we saw later, if a is negative, the effect is the same (stretching or compressing), but also the graph is reflected (“flipped”) over the x-axis.]

Next, we considered what would happen if we change the function by multiplying the input by a positive number a, in other words, we substitute ax for x in the formula for the function. Again we are starting with the graph of y = x^{3}+1, and we will consider the graph of y = f(ax) = (ax)^{3}+1 for various values of a which are positive. Notice again that the original function has a y-intercept at (o,1) and also that its graph passes through the points (-1, 0) and (1, 2). Pay attention to what happens to those points when we make changes. By using desmos to play around with different values for a which were positive, we found that this transformation stretches or compresses the graph toward the y-axis, depending on whether a is greater than 1 or a is between 0 and 1 .   The graph with sliders to change the value of a is here (in desmos). [As we saw later, if a is negative, the effect is the same (stretching or compressing), but also the graph is reflected (“flipped”) over the y-axis.]

We used the graphs linked in the last two paragraphs to see what would happen if we multiply the output or the input by -1.

Multiplying the output by -1 reflects the graph over the x-axis (because it changes the sign of all the y-values of the points on the graph).

Multiplying the input by -1 reflects the graph over the y-axis (because it changes the sign of all the x-values of the points on the graph).

After these examples we put some transformations together. See Example (5.7) in the textbook.

Note:  the “standard form” (sometimes called the “vertex form”) y=a(x-h)^{2} + k of the equation of a parabola can be explained using these transformations on the basic parabola y = x^{2}: the vertex moves to (h,k) and the factor a compresses or stretches the graph vertically, and if a is negative the graph opens downward (is “flipped”).
So these transformations are something you have actually experienced before!

In looking at the various transformations, it is extremely important to pay attention to whether you are acting on the output of the function (the value of the function) or the input to the function (the value of x which is being inputed). In other words, are we doing something to the formula for the function as a whole (the y-value), or are we substituting a different expression in place of the input x?
There is a very good discussion of the various transformations at the Regents Prep site.

• We then applied transformations and reading off of graphs to try and understand the “Test point” method of solving absolute value inequalities, for the example |x-5|\ge2.

We can find the graph of f(x) = |x-5| by transforming the basic graph y=|x|. Then we looked at the graph to answer the question posed by the inequality: for which values of x is f(x) \ge2? From the graph we can see why we only needed to test one x-value in each interval cut off by the solutions to the corresponding equation |x-5|=2, because in each interval the graph always lies either over the line y=2, or under the line y=2: the graph does not jump from over to under in the middle of those intervals! This only happens because the graph of our function is continuous (no breaks or jumps or gaps). We will return to this next time and in the future.

 

Homework:

• Review the examples of transformations that we discussed in class. Make sure that you understand how the transformations are moving or reshaping the graphs of the functions. [Plotting a few carefully selected points can help with this.]
• Do the following assigned problems from Session 5 from the Course outline: Exercises 5.1, 5.2, and 5.4. There is no WeBWorK for this topic.
• Make sure that you are familiar with all of the calculator techniques we have learned so far. Go back and do problem 4.3 especially, if you have not already done so. You will need it for Test 1!

• Don’t forget that Test 1 is scheduled for Wednesday. See the separate post for more information and review.

If you get stuck on any of the homework problems (routine or WeBWorK), don’t forget you can use the Piazza discussion board to ask questions!

Please bring your graphing calculator to class every time from now on! We will be practicing using it, and there is no substitute for hands-on experience.

 

 

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