1 / 23

Graphs and Functions: Understanding Coordinate Systems and Equations

Explore the fundamentals of graphs and functions, including rectangular coordinate systems, equations of lines, linear and quadratic functions, and exponential functions. Learn valuable concepts such as slopes, parallel and perpendicular lines, domain and range, and more.

laureni
Download Presentation

Graphs and Functions: Understanding Coordinate Systems and Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related