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ALGEBRA 1

Unit 2 Notes Chapter 3 Chapter 4 Chapter 5

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UPDATED FALL 2016

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Algebra 1 Section 3.3 Notes: Rate of Change and Slope Warm-Up Evaluate:

1) 2−43−1

2) 3−(−5)10−6

3) 6−73−(−3)

4) 6−(−6)10−1

Rate of Change: a that describes, on average, how much a quantity changes with respect to a change in another quantity.

Example 1: Use the table to find the rate of change. Then explain its meaning. In example 1 the rates of change have been constant. Many real-world situations involve rates of change that are _______ constant. Example 2: The graph below 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? A rate of change is constant for a function when the rate of change is the ____________ between any pair of points on the graph of the function. functions have a constant rate of change. Example 3: Determine whether each function is linear. a) b)

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Slope: the _____________ of the change in the y – coordinates (______________) to the change in the x – coordinates (_________) as you move from one point to another. Slope describes how ___________ a line is. The _____________ the absolute value of the slope, the ________________ the line. Because a linear function has a constant rate of change, __________________ on a nonvertical line can be used to determine its slope. The slope of a line can be ____________________, ___________________, _______________, or ______________________. If the line is not horizontal or vertical, then the slope is either positive or negative.

Example 4: Find the slope of a line that passes through each pair of points. a) (-3, 2) and (5, 5) b) (-3, -4) and (-2, -8) c) (-3, 4) and (4, 4) Example 5: Find the slope of the line that passes through (-2, -4) and (-2, 3).

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Section 3.3 DAY 2 Notes: Slope and Rate of Change Warm-up

1) Find the slope between the points (-5, 8) and (-10, - 3)

2) Write two points that a line passing through would have undefined slope.

3) Write two points that a line passing through would have zero slope.

Example 6: Find the value of r so the line passes through each pair of points and has the given slope. a) (6, 3), (r, 2); 𝑚 = 1

2 b) (–2, 6), (r, – 4); 𝑚 = −5 c) (r, –6), (5, –8); 𝑚 = −8

Example 7: Use the graph to approximate the average rate of change between x = - 1 and x = 3. Example 8: Which table shows golf scores that decrease by the same amount each game? A)

Game Score 1 92 2 88 3 82

B)

Game Score 1 72 2 69 3 65

C)

Game Score 1 79 2 76 3 73

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Algebra 1 Section 4.3 Notes: Writing Equations in Point-Slope Form Day 1 Warm-Up Find the slope of the line that has a x – intercept of – 2 and a y – intercept of 5. Point-slope form: an equation that can be written in the form __________________________where ___________ is the slope and ___________________ is a given point.

Example 1: Write an equation in point-slope form for the line that passes through (– 2, 0) with a slope of −3

2.

Example 2: Write an equation using point – slope form for each line using the given information. a.Slope is 1, Point is (4, 6) b. Slope is -4, Point is (-2, 8) c. Slope is 1

2, Point is (3, -2) d. Slope is 2

3, passes through (0, 2)

e. Slope is 0, Point is (2, -5) e. Slope is 1, passes through (-4, 0)

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Given Two Points: Step 1: _______________________________________________________. Step 2: Choose __________________________ of the points to use. Step 3: Follow the steps for writing an equation given the ____________________________________. Example 3: Write an equation using point – slope form for a line passing through the 2 points given. a. (- 3, 2) and (4, 7) b. (6, - 3) and (10, 1) Example 4: The figure shows trapezoid ABCD, with bases 𝐴𝐵���� and 𝐶𝐷���� Write an equation in point-slope form for the line containing the side 𝐵𝐶����.

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Algebra 1 Section 4.3 Notes: Writing Equations in Point-Slope Form Day 2 (Standard Form) Warm – Up Write an equation in point – slope form for the line with the given information. 1) (5, - 5), m = 2 2. ( - 1, 5), m = − 7

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Standard form: ______________________ where 𝐴 ≥ 0, A and B are not both zero, and A, B, and C are integers with a greatest common factor of 1. Example 1: Write the following equation in standard form. a) 𝑦 = 3

4𝑥 − 5 b) 𝑦 − 1 = 7(𝑥 + 5)

Writing an Equation in Standard Form Given Two Points Step 1: Find the __________________________________ between the two points. Step 2: Plug in the value of ______ and then choose either point to plug in as (x1, y1) into point – slope form. [__________________] Step 3: Rewrite the equation into standard form. [_________________________] Example 2: Write an equation in standard form that passes through the given points. a) ( - 4, 2) and (3, 8) b) x – intercept: (3, 0) and y – intercept (0, - 5) c) (−7,−3) and (−3, 5) d) (−1, 3) and y-intercept of 8. Example 3: Which equation has a graph with a slope of 2 and a y-intercept of 9? A. y - 9 = 2x B. y - 9 = -2x C. y = 2(x - 9) D. y = 2(x + 9)

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Algebra 1 Section 4.2 Notes: Writing Equations in Slope-Intercept Form Warm-Up The figure shows parallelogram ABCD. a) Write an equation in point – slope form of side BC. b) Write an equation in standard form of side BC. Write an equation in slope – intercept form given slope and point: Step 1: Plug in _____________________________________________________________________________ Step 2: Solve the equation for ________. Example 1: a) Write an equation of a line that passes through (2, – 3) with a slope of 1

2.

b) Write an equation of a line that passes through (–2, 5) with a slope of 3. If you are given two points through which a line passes, you can use them to find _________________. Then follow the steps above. Example 2: Write an equation of the line that passes through each pair of points. a) (–3, –4) and (–2, –8) b) (6, –2) and (3, 4) Constraint: a condition that a solution must ____________________. Equations can be viewed as constraints in a problem situation. The solutions of the equation meet the constraints of the problem.

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Example 3: During one year, Malik’s cost for self-serve regular gasoline was $3.20 on the first of June and $3.42 on the first of July. Write a linear equation to predict Malik’s cost of gasoline the first of any month during the year, using 1 to represent January. Linear extrapolation: the use of a linear equation to ________________ values that our outside the range of data. Example 4: On average, Malik uses 25 gallons of gasoline per month. He budgeted $100 for gasoline in October. Use the prediction equation in Example 3 to determine if Malik will have to add to his budget. Explain. Example 5: Which situation can be modeled by the equation y = 3x + 16? A) You have a summer job where you earn $3 per day cleaning cars, but earn an additional $16 per car cleaned during your shift. Let y equal your wages earned in one day. B) You have a summer job where you earn $16 per day cleaning cars but earn and additional $3 per car cleaned during your shift. Let y equal your wages earned in one day. C) You make a $3 payment on a new phone, and then will be $16 per month until the phone is paid off. Let y equal the total amount you will pay for the phone. D) You have $3 in your pocket, and want to buy a pizza that costs $16. Let y equal the total amount you need to buy the pizza. Example 6: The equation of a line is given in standard form. Find the slope of each line. a) 4x + 3y = 7 b) 3x + 2y = 10 Example 7: The equation C = 0.67t + 1.30 represents the cost C of a burger with t toppings. Which statement is true? A) Each topping costs $1.30. B) A burger with no topping costs $1.30. C) A burger with 3 toppings costs $1.97. D) A burger with no toppings costs $0.67.

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Algebra 1 Section 4.1 Notes: Graphing Equations in Slope-Intercept Form Slope-intercept form: an equation of the form ________________, where m is the ________ and b is the ______________. The variables m and b are called ________________ of the equation. Changing either value changes the equation’s graph.

When graphing an equation. 1) Plot the ______________________. 2) Use the slope to ____________ additional points. 3) _____________ a line through the points. Example 1: Write an equation in slope intercept form of the line with a slope of 1

4 and a y-intercept of – 1. Then graph the equation.

When an equation is not written in slope-intercept form, it may be easier to _________________ before graphing. Example 2: Graph 5𝑥 + 4𝑦 = 8

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Constant Functions: a linear function of the form _____________. Constant functions do not cross the ____________ except when 𝑦 = 0. Constant functions are ________________ lines therefore their slope is ________. Their domain is _________________, and their range is ___________. Vertical lines have _______________. So, equations of vertical lines __________________________________________. Example 3: a) Graph 𝑦 = −7 b) Graph x = 2 Writing an equation given a graph. 1) Locate the ________________________ 2) Find the slope by using ________________________ to find another point on the graph. 3) ___________________________________ in slope-intercept form Example 4: Write an equation in slope-intercept form for the line shown in the graph. Example 5: The ideal maximum heart rate for a 25 year old exercising to burn fat is 117 beats per minute. For every five years older than 25, that ideal rate drops three beats per minute. a) Write a linear equation to find the ideal maximum heart rate for anyone over 25 who is exercising to burn fat. b) Graph the equation. c) Find the ideal maximum heart rate for a 55 year old person exercising to burn fat.

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Example 6: The graph of a linear function is shown below. Which graph shows the same function with its y-intercept changed to 3?

A) B) C) D) Example 7: What is the equation of the line graphed? a) b)

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Algebra 1 Section 5.6 Notes: Graphing Inequalities in Two Variables Warm-Up Write an equation in slope-intercept form of each line. a) slope is 3; y – intercept is – 1 b) slope is −4

5; y – intercept is 0

Example 1: Determine which ordered pairs in the set are a part of the solution to the inequality. {(-2,-2), (1,-1), (1,1), (2,5), (6,0)} Intercepts x-intercept:

• Place where graph crosses _____________________. • Always in the form of _____________________. • Can be found by ________________________________________________________________.

y-intercept:

• Place where graph crosses _____________________. • Always in the form of _____________________. • Can be found by ________________________________________________________________.

Example 2: Find the x and y intercepts of 4 5 12x y+ = Graphing in Standard Form Steps for Graphing an Equation in Standard Form ( Ax By C+ = )

1. Find the _____________________________ by plugging in 0 for y and solving for x.

2. Find the _____________________________by plugging in 0 for x and solving for y.

3. Plot the two intercepts on the graph and connect the points to form a line. Example 3: Graph the equation 3 4 9x y+ =

5 7 10x y+ ≥

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Example 4: Which is the correct graph of 2x + 2y = -4? A) B) C) D) The graph of a linear inequality is the ____________________________ that represent _____________________________________ ___________________________________. An equation defines a ________________________, which divides the coordinate plane into _____________________________.

The boundary may or not be included in the solution. When it is included, the solution is a _________________________ (solid line). When not included, the solution is an _______________________ (dashed line).

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Example 5: Graph the inequality a) 2𝑦 − 4𝑥 > 6 b) 𝑥 − 1 > 𝑦 Example 6: Graph the inequality. a) 𝑥 + 4𝑦 ≥ 2 b) 2𝑥 + 3𝑦 ≥ 18

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You can use a coordinate plane to solve inequalities with one variable. Example 7: Use a graph to solve. a) 2𝑥 + 3 ≤ 7 b) −2𝑥 + 6 > 12 An inequality can be viewed as a constraint in a problem situation. Each solution of the inequality represents a combination that meets the constraint. In real-world problems, the domain and range are often restricted to nonnegative or whole numbers. Example 8: Write, Solve and Graph an Inequality a) Ranjan writes and edits short articles for a local newspaper. It takes him about an hour to write an article and about a half-hour to edit an article. If Ranjan works up to 8 hours a day, how many articles can he write and edit in one day?

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b) Neil wants to run a marathon at a pace of at least 6 miles per hour. Write and graph an inequality for the miles y he will run in x hours.

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Algebra1 1 Section 4.4 Notes: Parallel and Perpendicular Lines Warm – Up Which is the correct graph of x – y < 5? A) B) C) D) Parallel lines: lines in the same plane that .

Finding the equation of a line given the equation of a parallel line and a point on the line 1. Find the of the given line. 2. Substitute the provided and the from the given line into ______. Example 1: a) Write an equation in slope-intercept form for the line that passes through (4, – 2) and is parallel to 𝑦 = 1

2𝑥 − 7.

b) Write an equation in point-slope form for the line that passes through (4, – 1) and is parallel to 𝑦 = 1

4𝑥 + 7.

Perpendicular lines: lines that intersect at _______angles. The slopes of nonvertical perpendicular lines are .

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You can use slope to determine whether two lines are perpendicular.

You can determine whether the graphs of two linear equations are parallel or perpendicular by comparing the __________ of the lines. Example 2: a) Determine whether the graphs of 3𝑥 + 𝑦 = 12, 𝑦 = 1

3𝑥 + 2, and 2𝑥 − 6𝑦 = −5 are parallel or perpendicular. Explain.

b) Determine whether the graphs of 6𝑥 − 2𝑦 = −2, 𝑦 = 3𝑥 − 4, and 𝑦 = 4 are parallel or perpendicular. Explain. You can write the equation of a line perpendicular to a given line if you know a point on the line and the equation of the given line. Example 3: a) Write an equation in slope-intercept form for the line that passes through (4, – 1) and is perpendicular to the graph of 7𝑥 − 2𝑦 = 3. b) Write an equation in slope-intercept form for the line that passes through (4, 7) and is perpendicular to the graph of 𝑦 = 2

3𝑥 − 1.

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Algebra 1 Section 4.7 Notes: Inverse Linear Functions Warm-Up 1. Find the inverse of 𝑓(𝑥) = −3𝑥 + 4 2. Find the inverse of 𝑓(𝑥) = 1

2(𝑥 − 3)

Inverse relation: the set of ordered pairs obtained by ______________________________________of each ordered pair in a relation.

Notice the domain of a relation becomes the _____________________________________________, and the range of the relation becomes the ____________________________________________________. Example 1: Find the inverse of each relation. a) {(– 3, 26), (2, 11), (6, –1), (–1, 20)} b) The graphs of relations can be used to find and graph inverse relations. Example 2: Graph the inverse of each relation. a) b)

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Inverse function: A linear relation that is described by a function has an inverse function that can ___________________________ of the ________________________________. The inverse of the function 𝑓(𝑥) can be written as ____________ and is read 𝑓 of 𝑥 inverse or the inverse of 𝑓 of 𝑥.

Example 3: Find the inverse of each function. a) 𝑓(𝑥) = −3𝑥 + 27 b) 𝑓(𝑥) = 5

4𝑥 − 8

Example 4: Carter sells paper supplies and makes a base salary of $2200 each month. He also earns 5% commission on his total sales. His total earnings 𝑓(𝑥) for a month which he compiled 𝑥 dollars in total sales is 𝑓(𝑥) = 2200 + 0.05𝑥. a) Find the inverse function. b) What do 𝑥 and 𝑓−1(𝑥) represent in the context of the inverse function? c) Find Carter’s total sales for last month if his earnings for that month were $3450. Example 5: The table of values represents all points in the function f(x). Find the value of f-1(4).

x f(x)

-5 0

-2 2

0 4

1 -5

4 3

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Algebra 1 Section 5.3 Notes: Solving Multi-Step Inequalities Multi-step inequalities can be solved by _________________________________________in the same way you would solve a multi- step _________________________ Example 1: a) Adriana has a budget of $115 for faxes. The fax service she uses charges $25 to activate an account and $0.08 per page to send faxes. How many pages can Adriana fax and stay within her budget? Use the inequality 25 + 0.08𝑝 ≤ 115. Graph the solution. b) The Print Shop advertises a special to print 400 flyers for less than the competition. The price includes a $3.50 set-up fee. If the competition charges $35.50, what does the Print Shop charge for each flyer? When multiplying or dividing by a negative number, the __________________ of the inequality symbol ____________________. Example 2: Solve the inequality. a) 23 ≥ 10 − 2𝑤 b) 13 − 11𝑑 ≥ 79 c) 43 > −4𝑦 + 11 You can translate sentences into multi-step _________________________ and then solve them using the Properties of Inequalities. Example 3: Define a variable, write an inequality, and solve the problem. Then check your solution. a) Four times a number plus twelve is less than the number minus three. b) Two more than half of a number is greater than twenty-seven. When solving inequalities that contain ______________________________________, use the _______________________________ to remove the grouping symbols first. Then use the ______________________________________to simplify the resulting inequality.

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Example 4: Solve each inequality. Graph the solution on a number line. a) 6𝑐 + 3(2 − 𝑐) ≥ −2𝑐 + 1 b) 6(5𝑧 − 3) ≤ 36𝑧 c) 2(ℎ + 6) > −3(8 − ℎ) If solving an inequality results in a statement that is __________________, the solution set is the set of ________________________. This solution set is written as ____________________________. If solving an inequality results in a statement that is ____________, the solution set is the ______________________, which is written as the symbol ____. The empty set has no members. Example 5: Solve each inequality. Check your solution. a) −7(𝑘 + 4) + 11𝑘 ≥ 8𝑘 − 2(2𝑘 + 1) b) 2(4𝑟 + 3) ≤ 22 + 8(𝑟 − 2) c) 18 − 3(8𝑐 + 4) ≥ −6(4𝑐 − 1) d) 46 ≤ 8𝑚− 4(2𝑚 + 5)

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Example 6: A student solved the inequality as shown below. Determine if their solution is correct. If not, determine in which step the mistake was made, and identify the mistake. Given: −3(𝑥 − 4) + 6𝑥 − 2 ≥ 𝑥 − 4

Step 1: −3𝑥 + 12 + 6𝑥 − 2 ≥ 𝑥 − 4

Step 2: 3𝑥 + 10 ≥ 𝑥 − 4

Step 3: 4𝑥 + 10 ≥ −4

Step 4: 4𝑥 ≥ −14

Step 5: 𝑥 ≥ − 72

Example 7: Which statement is true for the inequality shown? 5(𝑥 − 1) + 2𝑥 > 7𝑥 − 5 A. The inequality is only true for numbers greater than 0 B. The inequality is only true for numbers less than 0 C. The inequality is never true D. The inequality is true for all values of x. Example 8: In the inequality 4𝑥 + 7 ≤ 451, x represents the number of T-shirts a printing company makes each day. Which statement MOST accurately describes how many T-shirts the company makes each day? A. Less than 111 T-shirts B. More than 111 T-shirts C. Exactly 111 T-shirts D. At most 111 T-shirts Example 9: Cyndy wants to buy some new make-up, but she cannot afford more than $35 before the sales tax is added. The eye shadow she wants are priced $8 each and the eye liner she wants are priced $5 each. Which inequality could be used to determine s, the number of eye shadows, and l, the number of eye liners Cyndy can afford? A. 8𝑠 + 51 ≥ 35 B. 8𝑠 + 51 ≤ 35 C. 5𝑠 + 81 ≥ 35 D. 5𝑠 + 81 ≤ 35

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Algebra 1 Section 5.5 Inequalities Involving Absolute Value Warm-Up Solve each inequality. 1. 𝑎

4< 16 2. 5

7𝑝 > −20

3. Define a variable and write an inequality for the phrase: One-half of Dan’s savings is less than $60. Solving Absolute Value Inequalities with Less Than When solving absolute value inequalities, there are two cases to consider. Case 1: The expression inside the absolute value symbols is ________________________. Case 2: The expression inside the absolute value symbols is ___________________. **The solution is the intersection of the solutions of these two cases. The inequality |𝑥| < 3 means that the distance between x and 0 is less than 3.

So, ____________________________________________________________. Example 1: Solve the inequality. Then graph the solution set. a) |𝑛 − 3| ≤ 12 b) |𝑥 + 6| < −8 c) |𝑥 + 6| < 8 Example 2:

a) The average annual rainfall in California for the last 100 years is 23 inches. However, the annual rainfall can differ by 10 inches from the 100 year average. What is the range of annual rainfall for California?

b) A recent survey showed that 65% of young adults watched online video clips. The margin of error was within 3% points. Find the range of young adults that watch video clips.

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Solving Absolute Value Inequalities with Greater Than Again, we must consider both cases. Case 1: The expression inside the absolute value symbols is ________________________. Case 2: The expression inside the absolute value symbols is _________________. The inequality |𝑥| > 3 means that the distance between x and 0 is greater than 3.

So, _____________________________________________________________________________. Example 3: Solve the inequality. Then graph the solution set. a) |3𝑦 − 3| > 9 b) |2𝑥 + 7| ≥ −11

c) |2𝑥 + 7| ≥ 11