1. Analyzing Graphs of Functions MATH 109 - Precalculus S. Rook

2. Overview Section 1.5 in the textbook: – Analyzing graphs – Finding zeros of a function – Increasing and decreasing functions – Even & odd functions – Application: Average rate of change 2

3. Analyzing Graphs

4. Given the graph of a function we should be able to: – State the domain and range Domain: the lowest and highest x-values Range: the lowest and highest y-values – Evaluate or estimate the value of the function (y) for any x Closed dots indicate a value is defined Open dots indicate a value is UNdefined 4

5. Analyzing Graphs (Example) Ex 1: Using the given graph of f(x), find the following if possible: a)f(8) b)f(4) c)f(3) d)Domain e)Range 5

6. Finding Zeros of a Function

7. Zeros of a Function Zeros of a function: all values x such that f(x) = 0 – i.e. x-intercepts – Recall that f(x) and y are interchangeable Being able to find zeros of a function is an important skill: – Facilitates sketching functions – Makes analyzing functions of a higher degree possible – Used in Calculus 7

8. Zeros of a Function (Example) Ex 2: Find the zeros of the function algebraically: a) b) c) h(x) = 2x 2 – x – 6 8

9. Increasing & Decreasing Functions

10. A function f is said to be: – Increasing if f(a) < f(b) for every value in an interval where a < b i.e. Rises from left to right on the interval (-9, -4) – Decreasing if f(a) > f(b) for every value in an interval where a < b i.e. Drops from left to right on the interval (3, 8) 10

11. Increasing & Decreasing Functions – Constant if f(a) = f(b) for every value in an interval where a < b Stays flat from left to right on the interval: (-4, 3) 11

12. Increasing & Decreasing Functions (Example) Ex 3: List the intervals where the graph of the given function: a)increases b)decreases c)is constant 12

13. Even & Odd Functions

14. A function f is an even function if f(-x) = f(x), for all x in the domain of f – Evaluating -x in f produces NO change – Symmetric about the y-axis A function f is an odd function if f(-x) = -f(x), for all x in the domain of f – Evaluating –x in f produces the OPPOSITE of f – Symmetric about the origin If a function f is neither even nor odd, it is said to be neither Why can’t a function be symmetric to the x-axis? 14

15. Even & Odd Functions (Example) Ex 4: Determine whether each function is even, odd, or neither and then describe the symmetry (if any): a) b) c) 15

16. Application: Average Rate of Change

17. Average Rate of Change Recall that in Section 1.3 one interpretation of slope is as a rate of change – In reference to linear equations We can also apply the concept to other functions: – Given 2 points on the graph of a non-linear function, the line connecting them is known as a secant line – The slope of the secant line is referred to as the average rate of change 17

18. Average Rate of Change (Continued) Recall that for a linear equation the formula for slope is The formula for average rate of change (denoted m sec ) is similar: – Adjusted for function notation f(x) and y are the same 18

19. Average Rate of Change (Example) Ex 5: Use the given function to find the average rate of change: a)f(x) = 2x 2 – 3x + 1, from x = -1 to x = 3 b)g(t) = 4t 3 – 20t, from t = 4 to t = 6 19

20. Summary After studying these slides, you should be able to: – Analyze a graph – Find the zeros of a function – State the intervals on which a function is decreasing, increasing, or stays constant – Demonstrate whether a function is even, odd, or neither – Evaluate the average rate of change for a non-linear function given endpoints of an interval Additional Practice – See the list of suggested problems for 1.5 Next lesson – A Library of Parent Functions (Section 1.6) 20