1. Chapter 8: Graphs and Functions

2. Rectangular Coordinate System 8.1

3. Rectangular Coordinate System 8.1

4. Distance and Midpoint Formulas 8.1

5. Circles 8.1

6. Lines and Slopes Equations of the form Ax + By = C can be visualized as a straight line Slope is rise/run x-intercept: set y = 0 y-intercept: set x = 0 8.2

7. Equations of Straight Lines Given the slope m and the y-intercept b, the slope-intercept form is y = mx + b Given a point (x 1,y 1 ) and the slope m, the point-slope form is y-y 1 = m(x-x 1 ) 8.2 & 8.3

8. Parallel and Perpendicular Parallel lines have the same slope Ex: y = 2x + 1 and y = 2x – 4 Perpendicular lines have slopes that are negative reciprocals Ex: y = 2x + 1 and y = -(1/2)x +3 8.2

9. Functions A relation is a set of ordered pairs A function is a relation in which for each value of the first component of the ordered pairs there is exactly one value of the second component Graph of a function obeys the vertical line test: any vertical line crosses at most once 8.4

10. Domain and Range When ordered pairs are of the form (x,y), x is the independent variable and y is the dependent variable The domain is the set of all values of the independent variable x The range is the set of all values of the dependent variable y 8.4

11. Linear Functions A function that can be written in the form f(x) = mx + b for real numbers m and b is a linear function. Example: cost and revenue models 8.4

12. Quadratic functions A function f is a quadratic function if f(x) = ax 2 + bx + c where a, b, and c are real numbers with a not equal to 0. 8.5

13. Graphing Quadratic Functions The graph of the quadratic function defined by f(x) = a(x-h) 2 + k, a not 0, is a parabola with vertex (h,k) and the vertical line x = h as axis of symmetry The graph opens up if a is positive and down if a is negative The graph is wide if |a| 1 compared to y = x 2 8.5

14. More Graphing Quadratics f(x) = ax 2 + bx + c 1.Decide if graph opens up or down 2.Find y-intercept by setting x = 0 3.Find x-intercept by solving f(x) = 0 4.Find vertex: x = -b/(2a) 5.Complete the graph 8.5

15. 8.5 #41 Steve has 100 meters of fencing material to enclose a rectangular exercise run for his dog. What width will give the enclosure the maximum area?

16. 8.5 #47

17. 8.6 Exponential Functions

18. Goes through (0,1) If b >1, then goes up from left to right If 0<b<1, then goes down from left to right x-axis is horizontal asymptote Domain is all numbers Range is y > 0

19. Compound Interest

20. Natural Exponent e

21. Logarithmic Functions

22. Exponential and Logarithmic Functions are Inverses

23. Useful properties