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Writing Linear Equations in Slope-Intercept Form Algebra 1

Exploration The graph at the right shows the cost of a smart phone plan.

1. What is the y intercept of the line? Interpret the y- intercept in the context of the problem.

2. Approximate the slope of the line. Interpret the slope in the context of the problem.

3. Write an equation that represents the cost as a function of data use.

4. Given the graph of a linear function, how can you write an equation of the line?

5. Given the slope of a linear function is 3 and y-intercept = -1. Can you write the equation of the line?

Writing Linear Equations in Slope-Intercept Form

6. Writing an Equation of a Line Identify the slope and y-intercept for each of the graphs below. Then write the equation for each line.

(a) (b)

7. Write an equation of the line that passes through the given points.

a) ( 0, -6) , ( 3, 9) b) ( 4, -1) , ( 0 , 2)

8. Write a linear function f with the given values.

a) f(0) = 8 and f(3) = -2 b) f(-4) = 5 and f(0) = -2

9. A Linear Model for Population The number of trout in a lake was estimated to be 44.5 thousand in 1995. During the next 10 years, the trout population increased by about 3 thousand per year.

(a) Write an equation to model the trout population (b) Graph the equation in part (a) for the years between 1995 and 2005.

Let t be the number of years since 1995. (c) Predict the trout population in 2002.

( 0, -1)

( -2, 0)

Writing Equations in Point–Slope Form and Slope Intercept form Algebra 1

Essential Question: How can you write an equation of a line when you are given the slope and a point on the line? Exploration: • Sketch the line that has the given slope and passes through the given point. • Find the y-intercept of the line. • Write an equation of the line.

Developing Point-Slope Form

1. Write an expression for the slope of the line using the points (2, 3) and (x, y).

m =

m = 4

(2, 3)

(x, y)

2. Given a point on the line and the slope, write each equation in point-slope form. (a) m = -5, (8, 2) (b) m = 8, (-3, 6)

(c) m = 43 , (-1, -5) (d) m = 1, (-4, 0)

( e ) Write an equation of the line shown in the graph

3. Given two points on the line, write each equation in point-slope form. (a) (1, 5), (-1, -5) (b) (-9, 10), (-4, -3) (c ) f(8) = -1 and f(6) = 0 4. Given a point on the line and the slope, write each equation in point-slope form. Then write the equation in slope-intercept form.

(a) m = -3, (-1, 5) (b) m = 21 , (10, -6)

Rewriting in Slope-Intercept Form

Sometimes it may be preferable to rewrite

your equation in slope-intercept form.

Example:

Point-slope form

Your equation is now in slope-intercept form.

5. Writing and Using a Linear Model (a) All the employees at Leo’s Coney Island are given a $0.40 per hour raise each year.

Shannon makes $7.15 per hour after three years as an employee. Write an equation that models her salary per hour S, in terms of n the number of years.

(b) Find her salary after 6 years of work.

6. To ride in a Royal Oak Taxi, you pay a start fee and a mileage fee. The table shows the total cost of a taxi ride for different distances. Can the situation be modeled by a linear equation? Explain. If possible, write a linear model that represents the cost as a function of the number of miles.

Distance ( miles) 4 8 10 12

Cost ( dollars) 13 23 28 33

Writing Equations of Parallel and Perpendicular Lines Algebra Essential Question: How can you recognize lines that are parallel or perpendicular? Exploration: Write each linear equation in slope-intercept form. Then use a graphing calculator to graph the three equations in the same square viewing window. Which two lines appear perpendicular?

1. a) 3x+ 4y = 6 2. a) 2x + 5y = 10 b) 3x – 4y = 12 b) -2x + y = 3 c) 4x – 3y = 12 c) 2.5x – y = 5

Two lines are perpendicular when ….

What did we learn about the equation of parallel lines in the last unit?

1.

(a) Write an equation of the line that is parallel to y = 3x + 4 and passes through the point (-2, 0) in slope- intercept form.

(b) Write an equation of the line that is parallel to y = -2x - 1 and passes through the point (2, 6) in point – slope form.

2. Determine which of the lines, if any, are parallel or perpendicular. Line a: 3y = 2x + 9 Line b: 3x + 2y = 8 Line c: 3y – 2x = -9

3. a) Give examples of numbers that are opposites: b) Give examples of numbers that are reciprocals: c) Give examples of numbers that are opposite reciprocals:

4. Write an equation of the line in slope-intercept form that passes through ( 4, -3) and is perpendicular to the line y = 4x - 1.

5. A road is constructed perpendicular to South Street. Write an equation that represents the new road.

South Street

( 5, 5)

( 11, 14)

Unit 5 Quiz Review Sheet Name ______________ Date _____

1. Write an equation in point-slope form of the line that passes through the given point and has the given slope.

a) ( 3, -4) b) ( -5, 2) c) ( ½ , 5) m = -2/3 m = -3 m = 4

2. Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form.

a) ( -5, 4) m = 2 b) ( -3, -6) , m = -4

3. Write an equation in point-slope form of the line that passes through the given points. Then write the equation in slope-intercept form. a) ( 2, 3), ( -4, 6)

b) ( -4, 18), ( 2, 9)

c) ( 0, -6) , ( 5, -6)

d) f(6) = 8 and f(3) = -2

4. Classified Ads : It costs $1.50 per day to place a one-line ad in the classifieds plus a flat

service fee. One day costs $3.50 and four days costs $8.00.

a) Write a linear equation that gives the cost in dollars, y, in terms of the number of days the ad appears, x.

b) Find the cost of a six-day ad.

5. Travel: You are flying from Houston to Chicago. You leave Houston at 7:30 A.M. At

8:35 A.M. you fly over Little Rock, a distance of 455 miles. a) Write a linear equation that gives the distance in miles, y, in terms of time, x. Let x represent the number of minutes since 7:30 A.M.

b)Approximately what time will you arrive in Chicago if it is 950 miles from Houston?

6. Find an equation of the line that is perpendicular to the line y = -2x + 6 and passes through ( -2, -4).

7. Find an equation of the line that is parallel to the line y = -3x + 2 and passes through ( 3, 4) .

8. Write the equation of the line.

9. Write an equation of a line that is perpendicular to 2x – y = 6.

10. Given a slope of - 5 and a y intercept of 7. Write the equation of the line.

a) b)

The Standard Form of a Linear Equation Algebra 1

We already know how to write the equation in 2 forms, what are they?

Exploration: You sold a total of $48 worth of cheese. You forgot how many pounds of each type of cheese you sold. Swiss cheese costs $8 per pound. Cheddar cheese costs $6 per pound.

1. Let x represent the number of pounds of Swiss cheese. Let y represent the number of pounds of cheddar cheese. Write a model that represents the situation.

2. Why would graphing using intercepts be a good way to graph this example?

3. Explain the meaning of the intercepts in the context of the problem.

4. Applications Involving Linear Equations in Standard Form

Linear Equations in standard form are often used in mathematical modeling, where equations are used to represent and solve problems that occur in real-life situations.

Example: You are in charge of buying hamburgers and hotdogs for our class cookout. The hamburgers cost $3 per pound and the hotdogs cost $1 per pound. You only have $30 to spend.

1. Model the possible combinations of hamburgers and hotdogs

Writing Linear Equations in Standard Form

We have learned how to write an equation in

To Write a Linear Equation In Standard Form

1. Given y =

2. Create integer coefficients by removing fractions

3. Isolate the variable terms on the left side and constant term on right

4. The equation in now in Standard Form Ax + By = C

5. . A Linear Model for Bird Seed Mixture You are buying $20 worth of birdseed that consists of two types of seed. Thistle seed, x, attracts finches and costs $4 per pound. Dark oil sunflower seed, y, attracts many kinds of songbirds and cost $2 per pound.

(a)Write an equation that represents the different (c) Graph the equation of the line using the table amounts of seed that you could buy.

(b) Complete this table

Thistle Seed (lb), x 0 1 2 3 4 5

Dark Oil Sunflower seed (lb), y

Are the following equations in standard form? If not put them in standard form.

6.

7. 2y = 3x – 5

8. 0.23x – 4.5y = 2.9 Reflection Question: Give an example of a horizontal line:____ Give an example of a vertical line: _____ Are these two equation written in standard form?______

Scatter Plots and Line of Best Fit Algebra

Essential Question: How can you use a scatter plot and line of fit to make conclusions about data?

1. Exploration: Collect data from 8 students and make a scatter plot.

Student Initials

Estimated number of minutes of exercise per day

Estimated number of

hours spent online per

day

EX: 45 min 2.5 hrs

2. Definition: A c___________________ is a measure of the strength of a r___________________

between two quantities.

Example: The more a student studies, the higher the student’s grades tend to be.

So, there is a c________________ between time spent studying and grades.

3. A trend line is a line on a scatter plot, drawn near the points, that shows a c_________________.

There should be about the same number of points above the line as below it.

4. Label each scatter plot as positive correlation, negative correlation, or no correlation.

5. Determining Correlation and Line of Best Fit

Determine whether each scatter plot has a positive correlation, negative correlation or no correlation. If the scatter plot shows a linear relationship, do your best to draw the line of best fit.

(a) (b) (c)

6. Drawing a Scatter Plot and Determining the Equation for the Line of Best Fit

(a) Draw a scatter plot of the given data (b) Does the data set show a positive correlation or negative correlation? (c) Do your best to draw the line of best fit. (d) Select two points on the line to determine the slope of the line: (e) Find the equation of the line of best fit.

(f) Use the equation of your trend line to estimate the body length of a 7-month-old panda

Section 5.4 Worksheet Name ___________________ Date _______

Bike Weights and Jump Heights 1. In BMX dirt-bike racing, jumping high or "getting air" depends on many factors: the rider's skill, the angle of the jump, and the weight of the bike. Here are data about the maximum height for various bike weights.

Use grid paper to plot the data (weight, height). If the data are linear, draw a trend or best-fit line.

2. Is there a positive, negative, or no relationship between bike weight and jump height? Explain your answer. 3. As the weight increases, the height ___________. 4. Find the slope or rate of change. What does this mean in words? 5. Predict the maximum height for a bike that weighs 21.5 pounds if all other factors are held constant.

Weight (pounds)

19 19 20 20 21 22 22 23 23.5 24

Height ( inches)

10.35 10.3 10.25 10.2 10.1 9.85 9.8 9.79 9.7 9.6

Predicting with Linear Models Algebra 1

Warm-Up Exercise Is a linear model appropriate for each graph? If so, is the line drawn on the scatter plot appropriate? If the line drawn is not appropriate, then draw in a line with a better fit. (a) (b) (c)

Writing Linear Models

When writing a linear equation to represent a data

set, follow the steps below.

- Make sure the data seems to follow a

_______________ relationship.

- Draw a ______ of _______ fit.

- Find two points on your line, and write the equation of

the line.

- Note: These do not have to be data points.

- If you have access to technology, then compute the

__________________________

1. Writing a Linear Model for a Line Drawn.

The scatter plot shows the average number of

hours per person per year that Americans spent

using the Internet since 1996. Write an equation

for the line drawn.

(a) Pick two points on the line that is drawn and find the line of best fit.

(b) Interpret the slope and y-intercept.

(c) Using technology we see that the least squares regression line is __________________________.

x 0 2 4 6

y 10 60 123 164

2. Predicting Using Linear Models

A child’s height (inches) from ages 2 through 12 is

given in the table below.

Age (x) 2 4 6 8 10 12

Ht (y) 34 39 48 56 63 67

(a) Make a scatter plot of the data.

(b) Draw a line of best fit and determine the equation

of the line.

(c) Using your equation, predict the child’s height at

age 5.

This method of prediction is called

______________________________. This is done

when making predictions within your data points.

In this case, making any prediction from 2-12 years

old. If the data set is linear, this method can be

accurate.

(d) Using your equation, predict the height at

age 40.

Does this prediction make sense?

This method of prediction is called

______________________________. This is

done when making predictions outside your

data points. In this case, making any

prediction below 2 or above 12 years old. We

do not know if the same pattern continues. The

pattern may not continue in the same linear

fashion. This is one of the dangers of

extrapolation.