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Chapter 1: Linear Functions, Equations, and Inequalities

Chapter 1: Linear Functions, Equations, and Inequalities. 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Linear Models 1.5 Linear Equations and Inequalities

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Chapter 1: Linear Functions, Equations, and Inequalities

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  1. Chapter 1: Linear Functions, Equations, and Inequalities 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Linear Models 1.5 Linear Equations and Inequalities 1.6 Applications of Linear Functions

  2. 1.3 Linear Functions • Linear Function • a and b are real numbers • Its graph is called a line • Its solution is an ordered pair, (x,y), that makes the equation true Example The points (0,6) and (–1,3) are solutions of since 6 = 3(0) + 6 and 3 = 3(–1) + 6.

  3. 1.3 Graphing a Line Using Points • Graphing the line Connect with a straight line. Plot the ordered pairs

  4. 1.3 Graphing a Line with the TI-83 • Graph the line with the TI-83 Xmin=-10, Xmax=10, Xscl=1 Ymin=-10, Ymax=10,Yscl=1

  5. 1.3 The x- and y-Intercepts, Zero of a Function • x-intercept: let y = 0 and solve for x • y-intercept: let x = 0 and solve for y • Zero of a function is any number c where f(c) = 0 • Two distinct points determine a line • e.g. (0,6) and (–2,0) are the y- and x-intercepts of the line y = 3x + 6, and x = –2 is the zero of the function.

  6. 1.3 Graphing a Line Using the Intercepts Example: Graph the line .

  7. 1.3 Application of Linear Functions A 100 gallon tank is initially full of water and being drained at a rate of 5 gallons per minute. a) What is the linear function that models this problem? b) How much water is in the tank after 4 minutes? c) Interpret the x- and y-intercepts.

  8. 1.3 Constant Function • Constant Function • b is a real number • the graph is a horizontal line • y-intercept: (0,b) • domain • range • Example:

  9. 1.3 Graphing with the TI-83 • Different views with the TI-83 • Comprehensive graph shows all intercepts

  10. 1.3 Slope In 1984, the average annual cost for tuition and fees at private four-year colleges was $5991. By 2004, this cost had increased to $20,082. The line graphed to the right is actually somewhat misleading, since it indicates that the increase in cost was the same from year to year. • Slope of a Line The average yearly cost was $705.

  11. 1.3 Formula for Slope • Slope m

  12. 1.3 Example: Finding Slope Given Points Determine the slope of a line passing through points (2, 1) and (5, 3).

  13. 1.3 Graph a Line Using Slope and a Point • Example using the slope and a point to graph a line • Graph the line that passes through (2,1) with slope

  14. 1.3 Slope of Horizontal and Vertical Lines • Slope of a horizontal line is 0 • Slope of a vertical line • Equation of a vertical line that passes through the point (a,b):

  15. 1.3 Slope-Intercept Form of a Line Slope-intercept form of the equation of a line • is the slope, and • b is the y-intercept

  16. 1.3 Matching Examples Solution: A. B. C. 1) C, 2) A, 3)B

  17. 1.3 Application of Slope • Interpreting Slope • In 1980, passengers traveled a total of 4.5 billion miles on Amtrak, and in 2000 they traveled 5.5 billion miles. • Find the slope m of the line passing through the points (1980, 4.5) and (2000, 5.5). Solution: b) Interpret the slope. Solution: Average number of miles people are traveling on Amtrak increased by around .05 billion, or 50 million miles per year.

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