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Section 2.1 Rectangular Coordinate Systems

1. Pythagorean Theorem

• In a right triangle, the lengths of the sides are related by the equation

where a and b are the lengths of the legs and c is the length of the hypotenuse.

• The converse is also true.

2. Distance Formula

The distance d between the points ( and ( is given by

3. The Midpoint Formula

The midpoint of the line segment joining the points ( and ( is given by

Example 1. Find all points on the y-axis that are a distance 8 from P(6,2).

Example 2. Find all points with coordinates of the form (a, a) that are a distance 8 from P(-3, 5).

212

2

12 yyxxd

2,

2

2121 yyxx

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Section 2.2 Graphs of equations

Ex1. What are the intercepts of the semicircle ?

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Theorem

The Standard Equation of a Circle with center (h,k) and radius r is given by

Ex2. Find an equation of the circle that has center C(-2,3) and contains the point D(4,5).

Ex3. Points P(-5,-1) and Q(3,4) are the endpoints of a diameter of a circle. Determine the

equation of the circle.

Ex4. Find the equation of the circle tangent to the x-axis, with center (5,8).

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To complete the square for the expression of the form , we add the square of the

half the coefficient of x . That means that we should use the following identity.

Proof: 1) Algebraic approach:

2) Geometric approach:

Ex6. Find the center and radius of the circle with equation

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G

Graphs of Basic Functions

Absolute Value Function

Domain

Range [0,

Squaring Function

Domain

Range [0,

Square Root Function

Domain

Range [0,

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Cubing Function

Domain

Range

Cube Root Function

Domain

Range

Half-Circle Function

Center (0, 0) Radius 1

Domain

Range

Reciprocal Function

Domain

Range

Horizontal Asymptote y = 0

Vertical Asymptote x = 0

Squared Reciprocal Function

Domain

Range

Horizontal Asymptote y = 0

Vertical Asymptote x = 0

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Ex. 7. Use tests for symmetry to determine which graphs from the list below are symmetric with

respect to the x-axis, the y-axis and the origin.

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2.3 Lines

1. Slope of the line passing through the points ),( 11 yx and ),( 22 yx is

.12

12

xx

yym

2. The equation of the line passing through the point ),( 11 yx with slope m is given by

)( 11 xxmyy

Point-slope formula

3. The equation of the line with slope m and y-intercept b is given by

bmxy

Slope-intercept formula

4. If line 1 is parallel to line 2, then .

If line 1 is perpendicular to line 2, then .

5. Slopes and equations for horizontal and vertical lines

6. How the slopes change.

Ex1. Find the equation of the line passing through two points (4,-2) and (-6,-5).

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Ex2. Find the equation of the line passing through the point (-3,0) and perpendicular to the line

7x+6y-6=0.

Ex.3 The y-intercept of the line show below is (0, 7). Find the slope of the line if the area of the

shaded region is 36 square units.

Ex. 5. 7 years ago a house was worth $83000. Now the house is worth $93000. Assume a linear

relationship between time and value,

(i) find a formula for the value , V(t), at time t. (t=0 refers to now).

(ii) What will be the value of the house in 4 years from now.

Ex. 6. A company purchases a piece of equipment for $20000. After 5 years, the piece of the

equipment loses 25% of its value. Assuming the value of the piece of the equipment is a linear

function of the time, determine the time (in years) it will take for the machine to be worth 35% of

its original value.

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Section 2.4: Definition of Function

Definition: A function from a set D to a set E is a correspondence that assigns to each element x

of D exactly one element y of E. The elements x in D are called inputs of the function and the

elements in E are called outputs of the function.

Functions can be expressed by different forms: Diagram, Table, Formula, Graph, Words.

Domain consist all possible inputs. It is a set of x values.

Range consist all possible outputs. It is a set of y values.

The graph of a function f is the graph of the equation y=f(x) for x in the domain of f.

Vertical line test: The graph of a set of points in a coordinate plane is the graph of a

function if every vertical line intersects the graph in at most one point.

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1. Find function values from formula and graph.

a. Replace the x in f(x) by a set of parentheses ( ).

b. Plug the input of the function in to ( ).

Example 1. Let Express the following functions in terms of x

(a)

(b)

Example 2.

2. Find domain of the function from formula.

Example 3. Find the domain of the following functions:

(a)

(b)

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(c)

(d)

3. Find domain and range of the function from graph.

4. The graphs of basic functions.

Example 4. Consider the function

(a) Sketch the graph of f(x)

(b) Find the domain and range for function f(x)

(c) Find the interval on which f is increasing or is decreasing, or is constant.

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5. Find and simplify a difference quotient.

Difference quotient

Example 5. Simplify the difference quotient for the following functions, if h is not zero.

(a)

(b)

6. Find a linear function.

Example 6. Let f(x) be a linear function such that f(3)=2 and f(5)=-7. Find f(x).

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7. Basic Geometry Formulas

Triangle

Area =

Circle

Area =

Circumference =

Trapezoid

Area =

Parallelogram

Area =

Rectangular Box

Volume =

Surface area =

Sphere

Volume =

Surface area =

Right Circular cylinder

Volume =

Lateral Surface area =

Right circular Cone

Volume =

8. Applications.

Four steps:

a. Identify the dependent variable of the problem.

b. Write the formula for the dependent variable.

c. Find relations between the independent variables.

d. Write the dependent variable as function of the desired independent variable.

Example 7. A rectangle has area 30. Express the perimeter P of the rectangle as a

function of the length x of the rectangle.

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Example 8. The point P(x, y) lies on the graph of . Express the perimeter of the

right triangle shown in the figure as a function of x.

Example 9. The figure shows a right circular cylinder with radius r and height h. The

surface area of the cylinder, including top and bottom, is 480 square feet. Express the

volume of the cylinder as a function of r.