Transcript Chapters 7 and 8 Slides

Chapter 7: Basic Concepts of
Algebra
Chapter 8: Graphs and Functions
7.1
Linear Equations
• An equation in the variable x is linear if it
can be written in the form
ax + b = c
where a,b,c are real numbers and a is not 0.
7.1
Helpful properties
• Addition property:
a = b and a + c = b + c
are equivalent (same solutions).
• Multiplication property:
a = b and ac = bc
Are equivalent as long as c is not 0.
7.1
Solving a linear equation
1.
2.
3.
4.
Clear fractions
Simplify each side separately
Isolate the variable terms on one side
Transform so that the coefficient of the
variable is 1
5. Check your solution
7.1
Kinds of Linear equations
• Conditional: finite number of solutions
Ex. 2x = 4
• Contradiction: no solutions
Ex. 2x + 1 = 2x +5
• Identity: true for any number
Ex. 2x + 2 = 2(x + 1)
7.2
Applications of Linear Equations
Key Words
Operation
Sum, more than, plus,
added, increased
Addition
Less than, difference,
minus, decreased
Subtraction
Times, multiplied by,
product
Multiplication
Quotient, ratio, divided
by
Division
7.2
Solving an Applied Problem
1. Read the problem carefully
2. Assign a variable to the unknown value,
and write down any other unknowns in
terms of this variable. Use tables,
diagrams, etc.
3. Write equation using the variable.
4. Solve the equation.
5. State the answer. Is it reasonable?
6. Check the answer in the words of the
original problem.
7.2
Examples
If a quotient of a number and
6 is added to twice the
number, the result is 8 less
than the number
7.2
Concert Revenue
U2 generated the top revenue on
the concert circuit in 2001. U2 and
second place *NSYNC together
took in $196.5 million in ticket
sales. If *NSYNC took in $22.9
million less than U2, how much did
each generate?
7.2
#40 Alcohol Mixture
How many liters of a 10%
alcohol solution must be
mixed with 40 liters of a 50%
solution to get a 40%
solution?
7.2
Coin Mixture
Dave collects US gold coins. He
has a collection of 80 coins. Some
are $10 coins and some are $20
coins. If the face value of the coins
is $1060, how many of each
denomination does he have?
7.2
#68 Time Traveled on a Visit
Steve leaves Nashville to visit his
cousin Dave in Napa, 80 miles away.
He travels at an average speed of 50
miles per hour. One half-hour later
Dave leaves to visit Steve, traveling at
an average speed of 60 miles per hour.
How long after Dave leaves will they
meet?
8.1
Rectangular Coordinate System
8.1
Rectangular Coordinate System
8.1
Distance and Midpoint Formulas
8.1
Circles
8.2
Lines and Slopes
• Equations of the form Ax + By = C
can be visualized as a straight
line
• Slope is rise/run
• x-intercept: set y = 0
• y-intercept: set x = 0
8.2 & 8.3
Equations of Straight Lines
• Given the slope m and the y-intercept b,
the slope-intercept form is
y = mx + b
• Given a point (x1,y1) and the slope m, the
point-slope form is
y-y1 = m(x-x1)
8.2
Parallel and Perpendicular
• Parallel lines have the same slope
Ex: y = 2x + 1 and y = 2x – 4
• Perpendicular lines have slopes that are
negative reciprocals
Ex: y = 2x + 1 and y = -(1/2)x +3
Functions
8.4
• A relation is a set of ordered pairs
• A function is a relation in which for
each value of the first component of
the ordered pairs there is exactly one
value of the second component
• Graph of a function obeys the
vertical line test: any vertical line
crosses at most once
8.4
Domain and Range
• When ordered pairs are of the form (x,y), x
is the independent variable and y is the
dependent variable
• The domain is the set of all values of the
independent variable x
• The range is the set of all values of the
dependent variable y
Linear Functions
•A function that can be written in
the form
f(x) = mx + b
for real numbers m and b is a
linear function.
•Example: cost and revenue models
8.4
7.7
Quadratic Equations
•An equation of the form
ax2 + bx + c = 0
where a,b,c are real numbers with a not
equal to 0, is a quadratic equation.
• Zero factor property (helpful for
factoring):
If AB = 0, then A = 0 or B = 0 or both
Quadratic Formula
7.7
Applications
The Toronto Dominion Centre in Winnipeg,
Manitoba is 407 feet high. Suppose that a
ball is projected upward from the top of the
centre and its position s in feet above the
ground is given by the equation s = -16t2 +
75t + 407, where t is the number of seconds
elapsed. How long will it take for the ball to
reach a height of 450 feet?
7.7 #68, page 355
• A club swimming pool is 30 feet wide by
40 feet long. The club members want a
border in a strip of uniform width around
the pool. They have enough material for
296 square feet. How wide can the strip
be?
Quadratic functions
8.5
•A function f is a quadratic function
if
f(x) = ax2 + bx + c
where a, b, and c are real numbers
with a not equal to 0.
8.5
Graphing Quadratic Functions
• The graph of the quadratic function
defined by f(x) = a(x-h)2 + k, a not 0, is a
parabola with vertex (h,k) and the vertical
line x = h as axis of symmetry
• The graph opens up if a is positive and
down if a is negative
• The graph is wide if |a|<1 and narrow if
|a|>1 compared to y = x2
More Graphing Quadratics
f(x) = ax2 + bx + c
1.Decide if graph opens up or down
2.Find y-intercept by setting x = 0
3.Find x-intercept by solving f(x) = 0
4.Find vertex: x = -b/(2a)
5.Complete the graph
8.5
8.5 #47
• Steve has 100 meters of fencing material
to enclose a rectangular exercise run for
his dog. What width will give the
enclosure the maximum area?
8.5 #50