1. Basic Properties of Functions

2. Things I need you to know about functions How to do basic substitution and recognize points How to graph a function. Sometimes without a calculator How to find the domain and range of a function How to find the zeros and y-intercept How to determine the intervals of increasing and decreasing How to identify asymptotes How to identify the basic behavior of the function just by looking at the equation

3. What is a function? The most common name is "f", but you can have other names like "g“ and basically they are all interchangeable with y.

4. Example 1 Example: with f(x) = x 2 : an x value of 4 4 2 = 16 so f(4) = 16 This means that when x = 4, f(x) = 16 Remember that f(x) = y Therefore when x = 4, y = 16 We now know a point on the graph (4, 16)

5. Domain and range Basically the domain is the list of all x values and the range is the list of all y values.

6. Zeros The zeros of a function f are found by solving the equation f(x) = 0. f(x) = -2 x + 4 f(x) = -2x + 4 = 0 x = 2

7. Example 2: Find the zeros of the quadratic function f(x) = -2x 2 -5x + 7 Factor the expression -2x 2 -5x+7 (-2x - 7)(x - 1) = 0 Therefore x = -7/2 and x=1

8. Basic Graphs of Functions A quadratic function is one of the form f(x) = ax 2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Ex: f(x) = x 2

9. Parabolas Parabolas have a maximum or minimum point based on whether they point up or down. This point is called the vertex.

10. Vertex of a Parabola The formula used to find the x value of the vertex is x = –b/2a which is taken from the quadratic formula x =

11. Vertex of a Parabola Find the vertex of the parabola given by the equation: f(x) = x 2 -4x+7 -b/2a =-(-4/2(1)), or 2 f(2)=2 2 -4(2)+7, or 3 The vertex of this parabola is (2, 3).

12. Increasing Functions A function is "increasing" if the y-value increases as the x-value increases, like this

13. Decreasing Functions A function is decreasing if the y-value decreases as the x-value increases:

14. Basic Graphs of Functions Absolute value of │x│ Domain (-∞,∞) Range: [0,∞)

15. Basic Graphs of Functions f(x) = √x Domain [0, ∞) Range [0, ∞)

16. Basic Graphs of Functions f(x) =√1-x 2 or (1-x 2 ) 1/2 Domain [-1, 1] Range [0, 1]

17. Asymptotes There are three types: horizontal, vertical and oblique:

18. Asymptotes An asymptote is a straight line that a graph comes closer and closer to but never touches.

19. Asymptotes The x-axis is a horizontal asymptote of the function y = 1/x. The graph and the x-axis come closer and closer but never touch. More precisely,

20. Example The graph of the function f(x) = 1/3x 3 + x 2 - 8x + 1 is shown here. For what values of x is the function f(x) = 1/3x 3 + x 2 - 8x + 1 decreasing? On (-4,2) – notice the interval is part of the domain or ‘x’ values!