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Linear Functions and Equations: Introduction, Examples, and Applications

This chapter covers the fundamentals of linear functions and equations, including graphing lines, finding intercepts, understanding slope, and applying linear functions in real-world situations.

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Linear Functions and Equations: Introduction, Examples, and Applications

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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 function f defined by where a and b are real numbers, is called a linear function. Its graph is called a line. Its solution is an ordered pair, (x,y), that makes the equation true.

  3. 1.3 Linear Functions Example The points (0,6) and (–1,3) are solutions of since 6 = 3(0) + 6 and 3 = 3(–1) + 6.

  4. 1.3 Graphing a Line Using Points Connect with a straight line. • Graphing the line • (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.

  5. 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

  6. 1.3 x- and y-Intercepts Locating x- and y-Intercepts To find the x-intercept of the graph of y = ax + b, let y = 0 and solve for x. To find the y-intercept of the graph of y = ax + b, let x = 0 and solve for y.

  7. 1.3 Zero of a Function Zero of a Function Let f be a function. Then any number c for which f (c) = 0 is called a zero of the function f.

  8. 1.3 Graphing a Line Using the Intercepts Example: Graph the line .

  9. 1.3 Application of Linear Functions A 100 gallon tank full of water is being drained at a rate of 5 gallons per minute. a) Write a formula for a linear function f that models the number of gallons of water in the tank after x minutes. b) How much water is in the tank after 4 minutes? c) Use the x- and y-intercepts to graph f. Interpret each intercept.

  10. 1.3 Constant Function • Constant Function b is a real number. The graph is a horizontal line. y-intercept: (0,b) Domain range Example:

  11. 1.3 Constant Function Constant Function A function defined by f(x) = b, where b is a real number, is called a constant function. Its graph is a horizontal line with y-intercept b. For b not equal to 0, it has no x-intercept. (Every constant function is also linear.)

  12. 1.3 Graphing with the TI-83 • Different views with the TI-83 • Comprehensive graph shows all intercepts

  13. 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 increase in cost (aka slope) was $705.

  14. 1.3 Formula for Slope The slope m of the line passing through the points (x1, y1) and (x2, y2) is

  15. 1.3 Example: Finding Slope Given Points Determine the slope of a line passing through points (2, 1) and (5, 3).

  16. 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

  17. 1.3 Slope of a Line Geometric Orientation Base on Slope For a line with slope m, • If m > 0, the line rises from left to right. • If m < 0, the line falls from left to right. • If m = 0, the line is horizontal.

  18. 1.3 Slope of Horizontal and Vertical Lines • The slope of a horizontal line is 0. • The slope of a vertical line is undefined. • The equation of a vertical line that passes through the point (a,b) is

  19. 1.3 Vertical Line Vertical Line A vertical line with x-intercept a has an equation of the form x = a. Its slope is undefined.

  20. 1.3 Slope-Intercept Form of a Line Slope-Intercept Form The slope-intercept form of the equation of a line is where m is the slope and b is the y-intercept.

  21. 1.3 Matching Examples Solution: A. B. C. 1) C, 2) A, 3)B

  22. 1.3 Application of Slope • Interpreting Slope In 1980, passengers traveled a total of 4.5 billion miles on Amtrak, and in 2007 they traveled 5.8 billion miles. a) Find the slope m of the line passing through the points (1980, 4.5) and (2007, 5.8). Solution: b) Interpret the slope. Solution: The average number of miles traveled on Amtrak increased by about 0.05 billion, or 50 million miles per year.

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