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Chapter 8: Graphs and Functions

Chapter 8: Graphs and Functions. 8.1. Rectangular Coordinate System. 8.1. Rectangular Coordinate System. 8.1. Distance and Midpoint Formulas. Circles. 8.1. Lines and Slopes. 8.2. Equations of the form Ax + By = C can be visualized as a straight line Slope is rise/run

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Chapter 8: Graphs and Functions

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  1. Chapter 8: Graphs and Functions

  2. 8.1 Rectangular Coordinate System

  3. 8.1 Rectangular Coordinate System

  4. 8.1 Distance and Midpoint Formulas

  5. Circles 8.1

  6. Lines and Slopes 8.2 • 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

  7. 8.2 & 8.3 Equations of Straight Lines • Given the slope m and the y-intercept b, the slope-intercept form is y = mx + b • Given a point (x1,y1) and the slope m, the point-slope form is y-y1 = m(x-x1)

  8. 8.2 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

  9. Functions 8.4 • 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

  10. Domain and Range 8.4 • 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

  11. Linear Functions 8.4 • 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

  12. Quadratic functions 8.5 • A function f is a quadratic function if f(x) = ax2 + bx + c where a, b, and c are real numbers with a not equal to 0.

  13. Graphing Quadratic Functions 8.5 • 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 and narrow if |a|>1 compared to y = x2

  14. More Graphing Quadratics 8.5 f(x) = ax2 + bx + c • Decide if graph opens up or down • Find y-intercept by setting x = 0 • Find x-intercept by solving f(x) = 0 • Find vertex: x = -b/(2a) • Complete the graph

  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

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