Toolkit Functions I

Tom Kelliher, MA 114

Sept. 26, 2001

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No new reading, but another online quiz on 2.3.

From Last Time

Characteristics of functions.

Introduction

Keys to understanding toolkit functions:

1. Behavior around the origin.

2. Above/below line y = x.

3. Symmetry: even/odd.

4. Asymptotes.

Functions and terminology:

1. Constant function.

   .

2. Linear function.

   Identity function.

3. Quadratic function.

   

   Parabola. Concave up. Even symmetry. Vertex --- turning point.

4. Cubic function.

   

   Steeper slope than quadratic. Odd symmetry. Concave up and down.

5. Square root function.

   Equivalent to .

   Calculator warning: Use parentheses --- .

   Principal (positive) square root --- otherwise, no function. But, we must remember when we have that , .

   Concave down.

6. Exponential function.

   

   Concave up. Horizontal asymptote at y = 0.

7. Reciprocal function.

   Equivalent to .

   Vertical asymptote at x = 0. Odd symmetry. Also has horizontal asymptote at y = 0.

8. Sine function

   

   Period is . Turning points at , , etc. Zeroes at 0, , , etc.

9. Absolute value function

   

   Finding the ABS key on the calculator --- three keys below 2nd key. Example of a piecewise-defined function:

   

   (How do I read one of these things???)

Class Practice

Use your calculator to graph each of the toolkit functions, using a window of and . For each function, answer these questions:

1. Identify the domain and range of the function.

2. For what part of the domain is the function above the line y = x? below?

3. What symmetry does the function have, if any? What are its asymptotes, if any?

Pg. 69: 1, 3.