Algebraic Reasoning, Graphing, and Connections with Geometry 8.1 Algebraic Expressions, Functions, and Equations 8.2 Graphing Points, Lines, and Elementary Functions 8.3 Connections Between Algebra and Geometry

8.1 Algebraic Expressions, Functions, and Equations

THREE IMPORTANT ROLES PLAYED BY ALGEBRA Algebra describes generality.

THREE IMPORTANT ROLES PLAYED BY ALGEBRA Algebra solves problems and explains patterns. Use a variable to represent an unknown value. Set up an equation which shows the relationship between the known and unknown values. Solve the equation.

THREE IMPORTANT ROLES PLAYED BY ALGEBRA Algebra and geometry of the Cartesian plane mix to form a problem-solving strategy. Graphs in the Cartesian plane can solve algebraic problems and algebra can solve geometric problems.

VARIABLES DESCRIBE GENERALIZED PROPERTIES Variables can be used to describe a general property or pattern. A generalized variable represents an arbitrary member of the set of elements for which the property or pattern holds.

VARIABLES EXPRESS RELATIONSHIPS Jolie was born on her three-year old sister Kendra’s birthday. Let J represent Jolie’s age and let K represent Kendra’s age.

VARIABLES SERVE AS UNKNOWNS IN EQUATIONS Elementary School: Middle School:

VARIABLES EXPRESS FORMULAS

DEFINITION: ALGEBRAIC EXPRESSION An algebraic expression in a mathematical expression involving variables, numbers, and operation symbols.

Example 8.2 Forming Algebraic Expression For each situation, form an algebraic expression that represents the requested values. a. The cost of every item is a store is increased by 15 cents. What is the cost of an item that used to cost c dollars? What is the old cost of an item that now costs d dollars? c + 0.15, d – 0.15 (in dollars) Slide 8-11 11

Example 8.2 continued For each situation, form an algebraic expression that represents the requested values. d. The electric power company charges $5 a month plus 7 cents per kilowatt-hour of electricity used. What is the monthly cost to use K kilowatt-hours? 5 + 0.07K (in dollars) Slide 8-12 12

DEFINITION: EQUATION An equation is a mathematical expression stating that two algebraic expressions have the same value. The equal sign, =, indicates that the expression on the left side has the same value as the expression on the right side of the symbol.

DEFINITION: SOLUTION SET OF AN EQUATION The solution set of an equation is the set of all values in the domain of the variables that satisfy the given equation. Two equations are equivalent if they have the same solution set.

Example We can solve the equation by multiplying through by . Try it. You should get to the following equation, which we solve to get solutions of However, the two equations are not equivalent. Why not?

DEFINITION: FUNCTION A function on a set D is a rule that associates to each element precisely one value y. The set D is called the domain of the function.

DEFINITION: RANGE OF A FUNCTION The range of a function f on a set D is the set of images of f. That is,

VISUALIZING FUNCTIONS: FUNCTIONS AS FORMULAS The area of a circle is a function of its radius and can be expressed as

VISUALIZING FUNCTIONS: FUNCTIONS AS TABLES The table gives the heights of three students. Note that there is no algebraic formula to relate the student to the height, but the relationship is established and is a function. STUDENT HEIGHT Athena 62” Bailey 68” Caroline 70”

VISUALIZING FUNCTIONS: FUNCTIONS AS ARROW DIAGRAMS

VISUALIZING FUNCTIONS: FUNCTIONS AS MACHINES

VISUALIZING FUNCTIONS: FUNCTIONS AS GRAPHS

Graphing Points, Lines, and Elementary Functions 8.2 Graphing Points, Lines, and Elementary Functions Slide 8-23

THE CARTESIAN COORDINATE PLANE

THE DISTANCE FORMULA

THE DISTANCE FORMULA Let P and Q be the points and Then the distance between P and Q is

THE DISTANCE FORMULA Find the distance between P and Q.

SLOPE OF A LINE SEGMENT OR LINE Let P (x1, y1) and Q (x2, y2), with x1 ≠ x2 be two points. Then the slope of the line segment or the line is given by

Example Find the slope of . 8 - 5 7 - 2

POINT-SLOPE FORM OF THE EQUATION OF A LINE The equation of the line through and having slope m is This is called the point-slope form of the equation of a line.

SLOPE-INTERCEPT FORM OF THE EQUATION OF A LINE The slope-intercept form of the equation of a line is where m is the slope and b is the y-intercept.

TWO-POINT FORM OF THE EQUATION OF A LINE The equation of a line through P(x1, y1) and Q(x2, y2), where x1 ≠ x2, is Slide 8-32 32

Connections Between Algebra and Geometry 8.3 Connections Between Algebra and Geometry Slide 8-33

Compute the lengths of the three sides. Example 8.13 Using Cartesian Coordinates to Prove That a Triangle is Isosceles Prove that the triangle with vertices R(1, 4), S(5,0), and T(7,6) is isosceles. Compute the lengths of the three sides. Since RT = ST, the triangle is isosceles. Slide 8-34 34

PARALLEL LINES The lines l and m in the Cartesian plane are parallel if they have no points in common or if they are equal. We write if l and m are parallel lines and if l and m are not parallel.

CONDITION FOR PARALLELISM Two lines in the plane are parallel if, and only if, both have the same slope or both are vertical lines. Slide 8-36 36

PERPENDICULAR LINES The lines l1 and l2 are perpendicular of they intersect at a 90° angle. Slide 8-37 37