AP Calculus BC 9/8 or 9/9 (Block Day) Section 1.2 Transformations and Inverses
What transformation maps the graph y = f(x) to the given function?
Example: y = f(x + h) where h > 0
Answer: Consider f(x) = x2 and f(x + 3) = (x + 3)2. Thus, f(x) is translated to the left 3 units and in general, by h units to the left.
1. y = f(x) + k 2. y = 2f(x) 3. y = - f(x)
4. y = f -1(x) 5. y = f(-x) 6. y = -f(-x)
You can use your nspire to explore parameters for questions like those above.
The updated version of the nspire’s software now includes “sliders.” Let’s practice using them.
Example
Step 1: Open a new problem, and create a graphs and geometry page.
Enter f1(x) = a * x2. Note that since a is not defined, no graph appears yet.
Step 2: Go to Menu > 1:Actions >A:Insert Slider. After you hit enter on the slider the parabola should appear.
Step 3: You should be able to move the slider itself by “grabby handing” it (hover over the slider and hold down the button at the center of the arrow keys) and then arrowing. If you would like to move the location of the slider, grabby hand the upper right hand corner of the slider box.
A Useful Note:
While pointing inside the slider, hit ctrl menu to access (1: Settings
2: Minimize 3: Animate 4: Delete). Adjust the settings to make your slider look like the one pictured here:
7. Use your nspire and two sliders to graph y = b·Sin(c·x). Describe the effect of each parameter b and c on the reference graph y=sin(x).
8. Review of Inverse Functions
A relation is a set of ordered pairs; the inverse of a relation is simply the same set with the x and y coordinates interchanged.
(a) To find the inverse of y = 3x – 1, switch x and y and solve for y.
Let f(x) = 3x – 1 Then f -1(x) = the expression you just solved for. Graph both of these functions. What geometric transformation relates them?
Give the point of intersection of the two graphs.
(b) Graph y = and its inverse (you’ll need to find the inverse). How are these graphs related? Give all points of intersection of these two graphs. What’s true about the points of intersection? Why does it make sense that inverses would intersect at such points (look at the definition of inverse above).
(c) Graph s(x) = and p(x) = x2. Students sometimes think that these graphs are inverses, but they are not. Name an ordered pair in p(x) that, when its coordinates are reversed, are not in s(x)
(d) We say that graphs of functions pass the vertical line test. What must be true about functions whose inverses will also be functions? These functions are called “one to one” functions. Explain this.