Name: Chapter 3: Linear Functions Page1 Lesson 3-1: Graphing Linear Equations Date: A equation is an equation that forms a line when it is graphed. Example 1: Identify Linear Equations A. Determine whether 5 + 3 = + 2 is a linear equation. Write the equation in standard form. B. Determine whether 3 4 =+8 is a linear equation. Write the equation in standard form. The x-coordinate of the point (, 0) at which the graph of an equation crosses the x-axis is an . The y-coordinate of the point (0, ) at which the graph of an equation crosses the y-axis is a . Standardized Test Example 2 Find the x- and y-intercepts of the segment graphed. A. B.
Transcript
Name: Chapter 3: Linear Functions
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Lesson 3-1: Graphing Linear Equations Date:
A equation is an equation that forms a line when it is graphed.
Example 1: Identify Linear Equations
A. Determine whether 5𝑥 + 3𝑦 = 𝑧 + 2 is a linear equation. Write the equation in standard form.
B. Determine whether 3
4𝑥 = 𝑦 + 8 is a linear equation. Write the equation in standard form.
The x-coordinate of the point (𝑥, 0) at which the graph of an equation crosses the x-axis is an .
The y-coordinate of the point (0, 𝑦) at which the graph of an equation crosses the y-axis is a .
Standardized Test Example 2
Find the x- and y-intercepts of the segment graphed.
A. B.
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Real-World Example 3: Find Intercepts
ANALYZE TABLES A box of peanuts is poured into bags at the rate of 4 ounces per second. The table shows the function relating to the weight of the peanuts in the box and the time in seconds the peanuts have been pouring out of the box.
A. Determine the x- and y-intercepts of the function.
B. Describe what the intercepts in the previous problem mean.
Example 4: Graph by Using Intercepts
Graph 4𝑥 – 𝑦 = 4 using the x-intercept and the y-intercept.
Step 1: To find the x-intercept, let 𝑦 = Step 2: To find the y-intercept, let 𝑥 =
Step 3: Graph
x
y
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Example 5: Graph by Making a Table
Graph 𝑦 = 2𝑥 + 2.
𝑥 𝑦 = 2𝑥 + 2 (𝑥, 𝑦)
−2
−1
0
1
2
x
y
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Lesson 3-2: Solving Linear Equations by Graphing Date:
The solution or of an equation is any value that makes the equation true. (when 𝑦 = 0)
Values of 𝑥 for which 𝑓(𝑥) = 0 are called .
Example 1: Solve an Equation with One Root
A. Solve 0 =1
2𝑥 + 3. B. Solve 2 =
1
3𝑥 + 3
Example 2: Solve an Equation with No Solution
A. Solve 2𝑥 + 5 = 2𝑥 + 3 B. 5𝑥 − 7 = 5𝑥 + 2
x
y
x
y
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Real-World Example 3: Estimate by Graphing
FUNDRAISING Kendra’s class is selling greeting cards to raise money for new soccer equipment. They
paid $115 for the cards, and they are selling each card for $1.75. The function 𝑦 = 1.75𝑥 – 115 represents
their profit 𝑦 for selling 𝑥 greeting cards. Find the zero of this function. Describe what this value means in this context.
x
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Lesson 3-3: Rate of Change and Slope Date:
Real-World Example 1: Find Rate of Change
DRIVING TIME Use the table to find the rate of change. Explain the meaning of the rate of change.
Real-World Example 2: Variable Rate of Change
TRAVEL The graph shows the number of U.S. passports issued in 2002, 2004, and 2006.
A. Find the rates of change for 2002–2004 and 2004–2006.
B. Explain the meaning of the rate of change in each case.
C. How are the different rates of change shown on the graph?
Example 3: Constant Rate of Change
Determine whether the function is linear. Explain.
A. B.
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Example 4: Positive, Negative, and Zero Slope
Find the slope of the line that passes through the given points.
A. (– 3, 2) and (5, 5) B. (– 3, – 4) and (– 2, – 8) C. (– 3, 4) and (4, 4)
Example 5: Undefined Slope
Find the slope of the line that passes through (– 2, – 4) and (– 2, 3).
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Example 6: Find Coordinates Given the Slope
A. Find the value of 𝑟 so that the line through (6, 3) and (𝑟, 2) has a slope of 1
2.
B. Find the value of 𝑝 so that the line through (𝑝, 4) and (3, – 1) has a slope of −5
8.
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Lesson 3-4: Direct Variation Date:
A is described by an equation of the form 𝑦 = 𝑘𝑥 or,
where 𝑘 is called the constant of variation, which is the same as the slope, so 𝑦 = 𝑚𝑥.
Example 1: Slope and Constant of Variation
Name the constant of variation for the equation that is graphed. Then find the slope of the line that passes through the pair of points.
A. B.
Example 2: Graph a Direct Variation
A. Graph 𝑦 = −3
2𝑥. B. Graph 𝑦 = 2𝑥
x
y
x
y
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Example 3: Write and Solve a Direct Variation Equation
A. Suppose 𝑦 varies directly as 𝑥, and 𝑦 = 9 when 𝑥 = – 3. Write a direct variation equation that relates 𝑥 and 𝑦.
Step 1: Find the value of 𝑘 (or 𝑚)
Step 2: Write the equation in the form 𝑦 = 𝑘𝑥.
B. Use the direct variation equation from part A to find 𝑥 when 𝑦 = 15.
Real-World Example 4: Estimate Using Direct Variation
TRAVEL The Ramirez family is driving cross-country on vacation. They drive 330 miles in 5.5 hours.
A. Write a direct variation equation to find the distance driven for any number of hours.
B. Graph the equation. C. Estimate how many hours it would take to drive 500 miles.
x
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Lesson 3-5: Arithmetic Sequences as Linear Functions Date:
Example 1: Identify Arithmetic Sequences
Determine whether the sequence is an arithmetic sequence. Explain.
A. – 15, – 13, – 11, – 9, . .. B. 7
8,
5
8,
1
8−
5
8….
Example 2: Find the Next Term
Find the next three terms of the arithmetic sequence – 8, – 11, – 14, – 17, …
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Example 3: Find the nth Term
A. Write an equation for the nth term of the arithmetic sequence 1, 10, 19, 28, … .
Step 1: Find the common difference.
Step 2: Write an equation.
B. Find the 12th term in the sequence.
C. Graph the first five terms of the sequence.
𝑛
1
2
3
4
5
D. Which term of the sequence is 172?
x
y
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Real-World Example 4: Arithmetic Sequences as Functions
NEWSPAPERS The arithmetic sequence 12, 23, 34, 45, . .. represents the total number of ounces that a bag weighs after each additional newspaper is added.
A. Write a function to represent this sequence.
Step 1: Find the common difference.
Step 2: Write the equation.
B. Graph the function and determine the domain.
x
y
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Lesson 3-6: Proportional and Nonproportional Relationships Date:
Real-World Example 1: proportional Relationships
ENERGY The table shows the number of miles driven for each hour of driving.
A. Graph the data. What can you deduce from the pattern about the relationship between the number of
hours of driving ℎ and the numbers of miles driven 𝑚?
B. Write an equation to describe this relationship.
C. Use this equation to predict the number of miles driven in 8 hours of driving.
x
y
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Example 2: Nonproportional Relationships
Write an equation in function notation for the graph.
