intermediate algebra 098a chapter 8 and section 3.6 more on functions and graphs
TRANSCRIPT
Jackie Joyner-Kersee - athlete
•“It is better to look ahead and prepare than to look back and regret.”
Function
• Is a relation in which no two ordered pairs have the same first components.
• {(1,2),(3,4)}
Vertical Line Test
• The graph of a relation represents a function if and only if no vertical line intersects the graph at more than one point
Functional Notation
• f(x) read “f of x”
• Name of the function is f
• x is the domain element
• f(x) is the value of the range
Repeated Evaluation of expression
• Enter expression in [Y=]
• [VARS][Y-VARS][1:Function][1:Y][ENTER]
Modeling
• Algebraic Models of situations are not perfect.
• Values of dates and variables need to be examined carefully
• Models can give predictions
• Some models are better than others
Intermediate Algebra 3.6
•****Odell’s Calculator Expectations
•Analysis of Functions
•Calculator Capabilities
Local Maximum
• Highest point in a “neighborhood”
• Local Minimum• Lowest point in a
“neighborhood.”
Points of Intersection
• The point(s) at which the two graphs of two function on the same set of axes intersect each other.
• Intermediate Algebra 098A 8.1• Graphing and Writing Linear Equations• Review of equations of Lines• Use of Graphing Calculator
Unknown author
• “Today, be aware of how you are spending your 1,440 beautiful moments, and spend them wisely.”
Def: Linear Equation
• A linear equation in two variables is an equation that can be written in standard form ax + by = c where a,b,c are real numbers and a and b are not both zero
Def: Intercepts
• y-intercept – a point where a graph intersects the y-axis.
• x-intercept is a point where a graph intersects the x-axis.
Procedure to find intercepts
• To find x-intercept• 1. Replace y with 0 in the given
equation.• 2. Solve for x• To find y-intercept• 1. Replace x with 0 in the given
equation.• 2. Solve for y
Horizontal line
• y = constant
• Slope is 0
• Examples: y = 5
• y = -3
• Can be done with calculator.
Vertical Line
• x=constant
• Undefined slope
• Examples:
• x =2
• x = -3
• Not graphed by calculator
Using Slope Intercept form to graph a line
• 1. Write the equation in form y=mx+b
• 2. Plot y intercept (0,b)
• 3. Write slope with numerator as positive or negative
• 3. Use slope – move up or down from y intercept and then right- plot point.
• 4. Draw line through two points.
Problem
• The percentage B of automobiles with airbags can be modeled by the linear function B(t)-5.6t –3.6, where t is the number of years since 1990.
• What is the slope of the graph of B?
• Answer is 5.6
Fred Couples – Professional Golfer
• “When you’re prepared you’re more confident: when you have a strategy you’re more comfortable.”
Def: Parallel Lines
• Two distinct non-vertical lines are parallel if and only if they have the same slope.
• Two distinct vertical lines are parallel.
Def 2: Perpendicular Lines
• The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
• If slope is a/b, slope of perpendicular line is –b/a.
y=mx+b
• Write the equation of a line given the slope and the y intercept.
• Line slope is 2 and y intercept (0,-3)
• y=2x-3
y=mx+b
• Write the equation of a line given the slope and one point.
• Slope of 2 and point (1,3)
• y=2x+1
Objective: Write equation of a line given the slope and one point
• Problem: slope of –3 through (2,-4)
• Answer: y=-3x+2
Objective – Write equation of line given two points
• Given points (-3,6),and (9,-2)
• Find slope
• Slope is –2/3
• Answer: y=(-2/3)x+4
Objective: Write equation of a line in slope-intercept form that
passes through (4,-1) and is parallel to y=(-1/2)x+3
•y=(-1/2)x+1
Section 8.4 – GayVariation and Problem Solving
• Direct Variation
• Inverse Variation
• Joint Variation
• Applications
Def: Direct Variation
• The value of y varies directly with the value of x if there is a constant k such that y = kx.
Procedure:Solving Variation Problems
• 1. Write the equation • Example y = kx• 2. Substitute the initial values and
find k.• 3. Substitute for k in the original
equation• 4. Solve for unknown using new
equation.
Example: Direct Variation
• y varies directly as x. If y = 18 when x = 5, find y when x = 8
• Answer: y = 28.8
Definition: Inverse Variation
• A quantity y varies inversely with x if there is a constant k such that
• y is inversely proportional to x.
• k is called the constant of variation.
ky
x
Procedure: Solving inverse variation problems
• 1. Write the equation• 2. Substitute the initial values
and find k• 3. Substitute for k in the
equation found in step 1.• 4. Solve for the unknown.