1. 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

2. 8.1
Algebraic Expressions, Functions, and Equations

3. THREE IMPORTANT ROLES PLAYED BY ALGEBRA
Algebra describes generality.

4. 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.

5. 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.

6. 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.

7. 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.

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

9. VARIABLES EXPRESS FORMULAS

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

11. 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 , d – 0.15 (in dollars)
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12. 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?
K (in dollars)
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13. 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.

14. 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.

15. 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?

16. 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.

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

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

19. 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”

20. VISUALIZING FUNCTIONS: FUNCTIONS AS ARROW DIAGRAMS

21. VISUALIZING FUNCTIONS: FUNCTIONS AS MACHINES

22. VISUALIZING FUNCTIONS: FUNCTIONS AS GRAPHS

23. Graphing Points, Lines, and Elementary Functions
8.2
Graphing Points, Lines, and Elementary Functions
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24. THE CARTESIAN COORDINATE PLANE

25. THE DISTANCE FORMULA

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

27. THE DISTANCE FORMULA
Find the distance between P and Q.

28. 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

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

30. 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.

31. 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.

32. 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
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33. Connections Between Algebra and Geometry
8.3
Connections Between Algebra and Geometry
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34. 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.
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35. 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.

36. CONDITION FOR PARALLELISM
Two lines in the plane are parallel if, and only if, both have the same slope or both are vertical lines.
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37. PERPENDICULAR LINES
The lines l1 and l2 are perpendicular of they intersect at a 90° angle.
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