Unit 8: Function Investigations

1. Unit 8: Function Investigations We have a TEST & PROJECT for this unit on 4/22 MA3A4. Students will investigate functions. a. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise. b. Investigate transformations of functions. c. Investigate characteristics of functions built through sum, difference, product, quotient, and composition.

2. Naming Functions • You name the function by the type of function it is. • For example, the function f(x) = x is the linear function.

3. Characteristics • We will go through all the characteristics of functions together today. • For the break you will complete a packet of functions and describe their characteristics. • Do ONE function at a time – graph it and look at the graph to help you determine as many characteristics possible. • The packet is due the same day your project is due.

4. Domain • All the x-values of a function • Graphically, it is read from left to right (smallest to largest) • Written as inequality or in interval notation Range • All the y-values of a function • Graphically it is read from bottom to top (smallest to largest) • Written as inequality or in interval notation

6. Extrema • These are local maximums or minimums • Functions can have one, both, or neither • It is where the function changes direction • Always written as a coordinate!!!

7. Increasing Interval • This is where the function has positive slope • It is written as inequality or interval notation • ALWAYS an interval of the domain • Graphically it is read from left to right.

8. Decreasing Interval • This is where the function has negative slope • It is written as inequality or interval notation • ALWAYS an interval of the domain • Graphically it is read from left to right.

9. Find the intervals of increase and decrease Increasing and decreasing are stated in terms of domain increasing (-1,2) decreasing increasing (1,-2)

10. Find the intervals of increase and decrease Increasing and decreasing are stated in terms of domain (0, 1) (2, 1) constant increasing decreasing

12. Asymptotes • A line that a graph gets closer and closer to, but never touches or crosses it. • Functions can have one, many, or no asymptotes • Asymptotes are written as excluded values of x • Can you think of a function that has an asymptote?

13. End Behavior • How the function behaves on the left and right side as x approaches negative or positive ∞ • Written “as x→-∞, f(x)→_____ and as x→+∞, f(x)→____” • A function can approach +∞, -∞, or some number • A function will always have end behavior!

14. Symmetry • Functions can have even symmetry, odd symmetry, or no symmetry • Functions can never be both even and odd • IT DOES NOT ONLY RELY ON THE DEGREE OF THE FUNCTION! • Graphically: • EVEN – end behavior is the same and it is centered on the y-axis • ODD – end behavior is opposite and it is centered on the origin

16. Symmetry • Algebraically: • EVEN – f(-x) = f(x) • ODD – f(-x) = -f(x)

17. Inverse • In your packet, you need to write the inverse function in this blank • Finding an inverse is easy: • Change f(x) to y • Solve the equation for x • Exchange x and y • If you are familiar enough with the functions then you can just use your brain ;)

18. For the remainder of class… • Complete the worksheet on determining function characteristics of the provided graphs • ALSO determine the symmetry of the functions • Start working on the parent function packet QUIZ NEXT CLASS