2
Graphs and
Functions
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1
2.2
Circles
• Center-Radius Form
• General Form
• An Application
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2
Circle-Radius Form
By definition, a circle is the set of all points
in a plane that lie a given distance from a
given point. The given distance is the
radius of the circle, and the given point is
the center.
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Center-Radius Form of the
Equation of a Circle
A circle with center (h, k) and radius r has
equation
which is the center-radius form of the
equation of the circle. A circle with center
(0, 0) and radius r has equation
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Example 1
FINDING THE CENTER-RADIUS
FORM
Find the center-radius form of the equation of
each circle described.
(a) center (– 3, 4), radius 6
Solution
Center-radius form
Substitute.
Watch
signs here.
Let (h, k) = (–3, 4) and r = 6.
Simplify.
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Example 1
FINDING THE CENTER-RADIUS
FORM
Find the center-radius form of the equation of
each circle described.
(b) center (0, 0), radius 3
Solution
The center is the origin and r = 3.
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Example 2
GRAPHING CIRCLES
Graph each circle discussed in Example 1.
(a)
Solution Writing the given equation in
center-radius form
gives (– 3, 4) as the
center and 6 as the
radius.
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Example 2
GRAPHING CIRCLES
Graph each circle discussed in Example 1.
(b)
Solution The
graph with center
(0, 0) and radius 3
is shown.
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General Form of the Equation
of a Circle
For some real numbers c, d, and e, the
equation
can have a graph that is a circle or a
point, or is nonexistent.
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General Form of the Equation of a
Circle
Consider
There are three possibilities for the graph
based on the value of c.
1. If c > 0, then r 2 = c, and the graph of the
equation is a circle with radius
2. If c = 0, then the graph of the equation is
the single point (h, k).
3. If c < 0, then no points satisfy the equation
and the graph is nonexistent.
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Example 3
FINDING THE CENTER AND RADIUS BY
COMPLETING THE SQUARE
Show that x2 – 6x + y2 +10y + 18 = 0 has a
circle as its graph. Find the center and radius.
Solution We complete the square twice,
once for x and once for y.
and
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Example 3
FINDING THE CENTER AND RADIUS BY
COMPLETING THE SQUARE
Add 9 and 25 on the left to complete the two
squares, and to compensate, add 9 and 25
on the right.
Add 9 and
25 on both
sides.
Complete the square.
Factor.
Because 42 = 16 and 13 > 0, the equation
represents a circle with center (3, – 5) and
radius 4.
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Example 4
FINDING THE CENTER AND RADIUS BY
COMPLETING THE SQUARE
Show that 2x2 + 2y2 – 6x +10y = 1 has a circle
as its graph. Find the center and radius.
Solution To complete the square, the
coefficients of the x2- and y2-terms must be 1.
Divide by 2.
Rearrange and
regroup terms.
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Example 4
FINDING THE CENTER AND RADIUS BY
COMPLETING THE SQUARE
Complete the square
for both x and y.
Factor and add.
Center-radius form
The equation has a circle with center at
and radius 3 as its graph.
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14
Example 5
DETERMINING WHETHER A GRAPH IS
A POINT OR NONEXISTENT
The graph of the equation
x2 + 10x + y2 – 4y + 33 = 0
is either a point or is nonexistent. Which is it?
Solution
Subtract 33.
and
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15
Example 5
DETERMINING WHETHER A GRAPH IS
A POINT OR NONEXISTENT
Complete the square.
Factor; add.
Since – 4 < 0, there are no ordered pairs (x, y),
with both x and y both real numbers, satisfying
the equation. The graph of the given equation
is nonexistent—it contains no points. (If the
constant on the right side were 0, the graph
would consist of the single point (– 5, 2).)
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16
Example 6
LOCATING THE EPICENTER OF
AN EARTHQUAKE
Suppose receiving stations A, B, and C are
located on a coordinate plane at the points
(1, 4), (–3, –1), and (5, 2). Let the distances from
the earthquake epicenter to these stations be 2
units, 5 units, and 4 units, respectively. Where on
the coordinate plane is the epicenter located?
Solution Graph the three circles. From the
graph it appears that the epicenter is located at
(1, 2). To check this algebraically, determine
the equation for each circle and substitute x = 1
and y = 2.
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Example 6
LOCATING THE EPICENTER OF
AN EARTHQUAKE
Station A:
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Example 6
LOCATING THE EPICENTER OF
AN EARTHQUAKE
Station B:
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Example 6
LOCATING THE EPICENTER OF
AN EARTHQUAKE
Station C:
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Example 6
LOCATING THE EPICENTER OF
AN EARTHQUAKE
The point (1, 2) lies on
all three graphs. Thus,
we can conclude that
the epicenter is at (1,
2).
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21
2
Graphs and
Functions
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1
2.1
Rectangular Coordinates and
Graphs
• Ordered Pairs
• The Rectangular Coordinate System
• The Distance Formula
• The Midpoint formula
• Equations in Two Variables
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2
Ordered Pairs
An ordered pair consists of two
components, written inside parentheses.
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3
Example 1
WRITING ORDERED PAIRS
Use the table to
write ordered
pairs to express
the relationship
between each
category and
the amount
spent on it.
Category
food
Amount
Spent
$8506
housing
$21,374
transportation
$12,153
health care
$4917
apparel and
services
$2076
entertainment
$3240
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WRITING ORDERED PAIRS
Example 1
(a) housing
Category
Solution
Use the data in the
second row:
(housing, $21,374).
food
Amount
Spent
$8506
housing
$21,374
transportation
$12,153
health care
$4917
apparel and
services
$2076
entertainment
$3240
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5
Example 1
WRITING ORDERED PAIRS
(b) entertainment
Category
Solution
Use the data in the last
row:
(entertainment, $3240).
food
Amount
Spent
$8506
housing
$21,374
transportation
$12,153
health care
$4917
apparel and
services
$2076
entertainment
$3240
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The Rectangular Coordinate
System
y-axis
Quadrant I
Quadrant II
P(a, b)
b
a
Quadrant III
x-axis
0
Quadrant IV
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The Distance Formula
Using the coordinates of ordered pairs, we
can find the distance between any two
y
points in a plane.
Horizontal side of
the triangle has
length
P(– 4, 3)
Q(8, 3)
x
R(8, – 2)
Definition of distance
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The Distance Formula
y
Vertical side of the
triangle has length
P(– 4, 3)
Q(8, 3)
x
R(8, – 2)
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The Distance Formula
y
By the Pythagorean
theorem, the length
of the remaining
side of the triangle is
P(– 4, 3)
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Q(8, 3)
x
R(8, – 2)
10
The Distance Formula
y
So the distance
between (– 4, 3) and
(8, – 2) is 13.
P(– 4, 3)
Q(8, 3)
x
R(8, – 2)
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The Distance Formula
To obtain a general formula for
the distance between two
points in a coordinate plane, let
P(x1, y1) and R(x2, y2) be any
two distinct points in a plane.
P(x1, y1)
Complete a triangle by
locating point Q with
coordinates (x2, y1). The
Pythagorean theorem gives
the distance between P and R.
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y
Q(x2, y1)
x
R(x2, y2)
12
Distance Formula
Suppose that P(x1, y1) and R(x2, y2)
are two points in a coordinate plane.
The distance between P and R, written
d(P, R) is given by the following
formula.
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Example 2
USING THE DISTANCE
FORMULA
Find the distance between P(– 8, 4) and
Q(3, – 2.)
Solution
Be careful when
subtracting a
negative number.
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Using the Distance Formula
If the sides a, b, and c of a triangle
satisfy a2 + b2 = c2, then the triangle
is a right triangle with legs having
lengths a and b and hypotenuse
having length c.
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Example 3
APPLYING THE DISTANCE FORMULA
Determine whether the points M(– 2, 5),
N(12, 3), and Q(10, – 11) the vertices of a
right triangle?
Solution A triangle with the three given
points as vertices is a right triangle if the
square of the length of the longest side
equals the sum of the squares of the lengths
of the other two sides.
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Example 3
APPLYING THE DISTANCE FORMULA
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Example 3
APPLYING THE DISTANCE FORMULA
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Example 3
APPLYING THE DISTANCE FORMULA
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Example 3
APPLYING THE DISTANCE FORMULA
The longest side, of length 20 units,
is chosen as the possible hypotenuse. Since
is true, the triangle
is a right triangle with
hypotenuse joining
M and Q.
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Collinear Points
We can tell if three points are collinear, that
is, if they lie on a straight line, using a similar
procedure.
Three points are collinear if the sum of the
distances between two pairs of points is
equal to the distance between the remaining
pair of points.
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Example 4
APPLYING THE DISTANCE
FORMULA
Determine whether the points P(– 1, 5),
Q(2, – 4), and R(4, – 10) are collinear.
Solution
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Example 4
APPLYING THE DISTANCE
FORMULA
Determine whether the points P(– 1, 5),
Q(2, – 4), and R(4, – 10) are collinear.
Solution
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Midpoint Formula
The coordinates of the midpoint M of
the line segment with endpoints P(x1,
y1) and Q(x2, y2) are given by the
following.
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Example 5
USING THE MIDPOINT FORMULA
Use the midpoint formula to do each of the
following.
(a)Find the coordinates of the midpoint M of
the line segment with endpoints (8, – 4) and
(– 6,1).
Solution
Substitute in the
midpoint formula.
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Example 5
USING THE MIDPOINT FORMULA
Use the midpoint formula to do each of the
following.
(b) Find the coordinates of the other endpoint
Q of a segment with one endpoint P(– 6, 12)
and midpoint M(8, – 2).
Solution
Let (x, y) represent the coordinates of Q. Use
the midpoint formula twice.
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Example 5
x-value of P
USING THE MIDPOINT FORMULA
x-value of M
x-value of P
y-value of M
Substitute
carefully.
The coordinates of endpoint Q are (22, – 16).
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Example 6
APPLYING THE MIDPOINT
FORMULA
The following figure depicts how a graph might
indicate the increase in the revenue generated
by fast-food restaurants in the United States
from $69.8 billion in 1990 to $195.1 billion in
2014. Use the midpoint formula and the two
given points to estimate the revenue from fastfood restaurants in 2002, and compare it to the
actual figure of $138.3 billion.
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Example 6
APPLYING THE MIDPOINT
FORMULA
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Example 6
APPLYING THE MIDPOINT
FORMULA
Solution The year 2002 lies halfway
between 1990 and 2014, so we must find
the coordinates of the midpoint of the
segment that has endpoints
(1990, 69.8) and (2014, 195.1).
Thus, our estimate is $132.5 billion, which
is less than the actual figure of $138.3
billion.
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Example 7
FINDING ORDERED-PAIR
SOLUTIONS OF EQUATIONS
For each equation, find at least three ordered
pairs that are solutions
(a)
Solution Choose any real number for x or y
and substitute in the equation to get the
corresponding value of the other variable.
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Example 7
FINDING ORDERED-PAIR
SOLUTIONS OF EQUATIONS
(a)
Solution
Let x = – 2.
Let y = 3.
Multiply.
Add 1.
Divide by 4.
This gives the ordered pairs (– 2, – 9) and (1, 3).
Verify that the ordered pair (0, –1) is also a
solution.
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Example 7
(b)
Solution
FINDING ORDERED-PAIR
SOLUTIONS OF EQUATIONS
Given equation
Let x = 1.
Square each side.
Add 1.
One ordered pair is (1, 2). Verify that the
ordered pairs (0, 1) and (2, 5) are also
solutions of the equation.
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Example 7
FINDING ORDERED-PAIR
SOLUTIONS OF EQUATIONS
(c) A table provides an organized method for
determining ordered pairs.
x
y
Solution
–2
0
–1
–3
0
–4
1
–3
2
0
Five ordered pairs are (–2, 0), (–1, –3), (0, –4),
(1, –3), and (2, 0).
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Graphing an Equation by
Point Plotting
Step 1 Find the intercepts.
Step 2 Find as many additional ordered
pairs as needed.
Step 3 Plot the ordered pairs from Steps 1
and 2.
Step 4 Join the points from Step 3 with a
smooth line or curve.
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GRAPHING EQUATIONS
Example 8
(a) Graph the equation
Solution
Step 1 Let y = 0 to find the x-intercept, and let
x = 0 to find the y-intercept.
The intercepts are
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Example 8
GRAPHING EQUATIONS
(a) Graph the equation
Solution
Step 2 We find some other ordered pairs
(also found in Example 7(a)).
Let x = – 2.
Let y = 3.
Multiply.
Add 1.
Divide by 4.
This gives the ordered pairs (– 2, – 9) and (1, 3).
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GRAPHING EQUATIONS
Example 8
(a) Graph the equation
Solution
Step 3 Plot the four ordered pairs from
Steps 1 and 2.
Step 4 Join the
points with a
straight line.
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Example 8
GRAPHING EQUATIONS
(b) Graph the equation
Solution
Plot the ordered pairs
found in Example 7b,
and then join the points
with a smooth curve. To
confirm the direction the
curve will take as x
increases, we find
another solution, (3, 10).
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Example 8
GRAPHING EQUATIONS
(c) Graph the equation
Solution Plot the points and join them with a
smooth curve.
x
y
–2 0
–1 –3
0 –4
1 –3
2
0
This curve is called a parabola.
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Wk2 Applications
Pick one concept presented this week. Discuss one real-life scenario in which
this concept is applicable. In other words, state one way in which you can use
this concept in every day life.
8
Wk2 Reflections
lease share a summary of the week's activity. The following are
things you can consider to include in your reflection:
1. Summarize what you learned.
2. Identify problem areas.
3. Discuss strengths. What you like about class?
4. Discuss weaknesses. What you dislike about class?
5. Discuss ways to improve class. What can you do to improve your
performance? What can your instructor do to improve your performance?
1. I learned how to solve for range, domain, radius, graphing, how to read
graphs, and how to determine if something is a function or not.
2. I had a little bit of trouble in each section, but once I keep doing the
problems over again I figured it out.
3. I like that I can keep working the problems over and over again until I g
to know them and learn them.
4. What i dislike about the class is not having someone to show me how to
work these problems.
5. I can Improve my performance by doing more practice problems and
asking someone for help.

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