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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.