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Introduction to Linear Functions and Equations

This chapter introduces the concept of linear functions, equations, and inequalities. It covers topics such as real numbers, the rectangular coordinate system, equations of lines, distance and midpoint formulas, and applications of linear functions.

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Introduction to Linear Functions and Equations

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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. 1.1 Sets of Real Numbers Sets of Real Numbers: • Natural Numbers: • Whole Numbers: • Integers: • Rational Numbers: • Irrational Numbers:

  3. 1.1 Example Indicate the set each number belongs to.

  4. 1.1 The Set of Real Numbers and the Number Line • Real Numbers: • Every real number corresponds to a point on the number line. -4 -3 -2 -1 0 1 2 3 4

  5. 1.1 The Rectangular Coordinate System • The number corresponding to a particular point on the number line is called the coordinate of the point. • This correspondence is called a coordinate system.

  6. P(a, b) 1.1 The Coordinate Plane • Cartesian Coordinate System • xy-plane (or coordinate plane) Quadrant I Quadrant II Origin Quadrant III Quadrant IV

  7. 1.1 The TI-83 Viewing Window • Limitations in portraying coordinate systems on the calculator screen 1. Resolution 2. Scaling Xmin=-60, Xmax=60, Xscl=1 Xmin=-60, Xmax=60, Xscl=10 Ymin=-40, Ymax=40, Yscl=1 Ymin=-40, Ymax=40,Yscl=10

  8. 1.1 Rounding Numbers Mode Setting Display

  9. 1.1 Roots • Calculators have the ability to express numbers like: • Other special keys:

  10. 1.1 The Pythagorean Theorem Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

  11. 1.1 The Distance Formula d Q(x1,y2)

  12. 1.1 The Distance Formula Distance Formula Suppose that P(x1, y1) and R(x2, y2) are two points in a coordinate plane. Then the distance between P and R, written d(P,R), is given by the distance formula

  13. 1.1 Example Using the Distance Formula • Find the length of the line segment that joins the points P(8, 4) and Q(3, 2). Solution:

  14. 1.1 Midpoint Formula • The midpoint of the line segment with endpoints and has coordinates

  15. 1.1 Midpoint Formula Example • Find the midpoint M of the segment with endpoints (8, 4) and (9,6). Solution:

  16. 1.1 Application: Estimating Tuition and Fees • In 2002, average tuition and fees at public colleges and universities were $4081, whereas they were $6585 in 2008. Use the midpoint formula to estimate tuition and fees in 2005. Compare your estimate with the actual value of $5491. The year 2005 lies midway between 2002 and 2008. Therefore, we can use the midpoint formula. The midpoint formula estimates tuition and fees at public colleges and universities to be $5333 in 2005. This is within $158 of the actual value.

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