Chapter 2: Linear Equations and Functions
Chapter 2.1: Functions and Their Graphs
Relation:
Domain:
Range:
Function:
Example: Identify the domain and range. Then tell whether the relation is a function.
Ordered Pairs:
Coordinate Plane:
Example: Graph the relations from the first example.
Vertical Line Test for Functions:
Example: The graph shows the average daily temperature (°F) and number of visitors
for the beach. Is the number of visitors a function of the daily temperature? Explain.
Evaluating Functions:
General Graphing of Functions:
Example: Graph y = x – 2
Linear Function:
Function Notation:
Example: Decide whether the function is linear. Then evaluate the function when x = 3.
f(x) = x2 + 4x – 1
g(x) = -3x + 4
Example: At 2.4 calories burned per pound of weight each hour, the calories c burned in h hours by a 110 pound
person walking briskly can be modeled by c = 110(2.4)h.
For a walk of up to 4 hours, identify the domain and range.
Graph the function. Use the graph to estimate how long it takes to burn 400 calories.
Chapter 2.2: Slope and Rate of Change
Slope:
Example: Find the slope of the line passing through (-2, -4) and (3, -1).
Classifications of Slope
Example: Find the slope of the line through the given points. Then tell whether the line through the given points
rises, falls, is horizontal, or is vertical.
(-2, 3), (1, 5)
(1, -2), (3, -2)
Example: Tell which line is steeper
Line 1: through (1, -4) and (5, 2)
Line 2: through (-2, -5) and (1, -2)
Slopes:
Example: Tell whether the linear are parallel, perpendicular, or neither.
Line 1: through (1, -2) and (3, -2)
Line 1: through (-2, -2) and (4, 1)
Line 2: through (-5, 4) and (0, 4)
Line 2: through (-3, -3) and (1, 5)
Example: The slope of a road, or grade, is usually expressed as a percent. For example, if a road has a grade of 3%, it
rises 3 feet for every 100 feet of horizontal distance.
Find the grade of a road that rises 75 ft over a horizontal distance of 2000 feet.
Find the horizontal length x of a road with a grade of 4% if the road rises 50 feet over its length.
Example: The average local monthly US cell phone bill decreased from $61.48 in 1993 to $47.70 in 1996. Find the
average rate of change and use it to estimate the average monthly bill in 1997.
Chapter 2.3: Quick Graphs of Linear Equations
x-intercept:
y-intercept: point
Slope-Intercept form:
Example: Graph
=
+1
Example: To buy a $1200 stereo, you pay a $200 deposit and then make weekly payments according to the equation
a = 1000 – 40t, where a is the amount you owe and t is the number of week.
How much do you owe originally?
What is your weekly payment?
Graph the model.
Standard Form:
Example: Graph 3x – 2y = 6 using standard form. Then rewrite the equation in
slope-intercept form.
Horizontal Lines:
Vertical Lines:
Example:
Graph y = -2
Graph x = 3
Chapter 2.4: Writing Equations of Lines
Writing Equations of Lines:
Given slope and y-intercept
Given a slope and a point
Given two points
Example: Write the equation of the line shown.
Example: Write an equation of the line that passes through (-3, 4) and has a slope of 2/3.
Example: Write an equation of the line that passes through (2, -3) and is (a) perpendicular to and (b) parallel to the
line of y = 2x – 3.
Example: Write an equation of the line that passes through (1, 5) and (4, 2).
Example: In 1984, Americans purchased an average of 113 meals or snacks per person at restaurants. By 1996, this
number was 131. Write a linear model for the number of meals or snacks purchased per person annually. Then use
the model to predict the number of meals or snacks that will be purchased per person in 2006.
Direct Variation:
Example: The variables x and y vary directly, and y = 15 when x = 3.
Write and graph an equation relating x and y.
Find y when x = 9.
Example: Tell whether the data show direct variation. Is so, write an equation relating x and y.
(3, 12), (6, 24), (9, 38), (12, 50), (15, 65), where (x, y) = (Hours worked, Wages ($))
(8, 2.40), (16, 4.80), (24, 7.20), (32, 9.60), (40, 12.00), where (x, y) = (weight of nuts (oz), price ($)).
Chapter 2.5: Correlation and Best-Fitting Lines
Scatter Plots:
Example: Describe the correlation shown by the scatter plot.
Line of Best Fit:
Example: The data pairs give the average speed of an airplane
during the first 10 minutes of a flight, with x in minutes and y in
miles per hour. Approximate the best-fitting line for the data.
(1, 180), (2, 250), (3, 290), (4, 310), (5, 400), (6, 420), (7, 410), (8,
490), (9, 520), (10, 510)
Chapter 2.6: Linear Inequalities in Two Variables
Linear Inequalities:
Example: Check whether the ordered pair is a solution of 4x – 2y ≥ 8.
(3, 3)
(-2, -9)
Example:
Graph y ≥ 2
Graph x > -1
Example: You have $200 to spend on CDs and music videos. CDs cost $10 each
and music videos cost $15.
Write a linear inequality in two variables to represent the number of
CDs x and music videos y you can buy.
Graph the inequality. Discuss three possible solutions for the real-life
situation.
Graph 4x + 2y ≥ 8
Chapter 2.7: Piecewise Functions
Piecewise Functions
Example: Evaluate f(x) when ( ) =
3 + 2, ≤ 3
− 1, > 3
x=0
x=3
x=6
Example: Graph this function ( ) =
+ , > 2
− + 1, ≤ 2
1,
2,
Example: Graph this function ( ) =
3,
4,
− 4 ≤ < −3
− 3 ≤ < −2
− 2 ≤ < −1
−1≤ <0
Step Function:
Example: Write equations for the piecewise function whose graph is shown.
Example: Shipping costs $6 on purchases up to $50, $8 on purchases over $50 and up to $100, and $10 on purchases
over $100 up to $200. Write a piecewise function for these charges. Give the domain and range.
Example: A plane descends from 5000 ft at 250 ft/min for 6 min. Over the next 8 min, it descends at 150 ft/min.
Write a piecewise function for the altitude A in terms of the time t. What is the plane’s altitude after 12 min?
Chapter 2.8: Absolute Value Functions
Absolute Value Functions:
Example: Graph y = -|x – 1| + 1
Example: Write an equation of the graph shown.
Example: The front of a roof with its outer edges 8 feet above the ground can be modeled by the following equation,
with x and y both in feet. Graph the function. Interpret the domain and range in this context.
= − | − 9| + 14
Example: You want to shoot the eight ball into the corner pocket on a pool table 10 feet long and 5 feet wide. The
ball is at (2, 1); the pocket is at (10, 0). You plan to bank off the side at (6, 5).
Write an equation for the path of the ball.
Do you make your shot?