Module 1: LINES
Exploring Calculus

Part I--SLOPE
 

We begin investigating lines by looking at the concept of the slope of a line.

  Display the graph with four lines.     
 
 

Critical Thinking Questions
 

1. How could you find the slope of each line graphed on the screen?Click here to go to the hint.Hint
 

2. Consider the line passing through the points (-2, 4) and (3, -5) .

    a. What is the slope of the line?

    b. Do the y-values increase or decrease as x increases?
 

3. The slope of a line is - .

    a. Is the line increasing or decreasing?

    b. If x decreases by 2 units, what is the corresponding change in y?

    c. What is the change in x, if y changes by -6?
 

4. Suppose a line has slope - and passes through (5, 4). Will it also pass through  (-5, 0)?
    Show why or why not.
 

5. Pick any two points on a vertical line.

    a. What do the two points have in common?

    b. If the line passes through the point (a, b), what equation do you think describes this type of line?
 

6. Now, pick any two points on a horizontal line.

    a. What do the two points have in common?

    b. If the line passes through the point (a, b), what equation do you think describes this type of line?
 

7. When a fraction has:

    a. zero numerator and non-zero denominator, what is the value of the fraction?

    b. zero denominator and non-zero numerator, what is the value of the fraction?
 
 

Skill Exercises
 
 

1. Find the slope of each of the four lines graphed on the screen.

    m1 ____________ m3 _______________

    m2 ____________ m4 _______________

2. Rising lines have a ____________   slope. Falling lines have a ____________  slope.
 

3. A line increasing at a 45-degree angle has a slope of _______________.  A line decreasing at a 45-degree angle has a slope of _______________.  A horizontal line has _______________.  A vertical line has _______________slope.
 

4. This question is about parallel lines.

    a. How are the slopes of parallel lines related?

    b. Determine the number  a such that the line passing through (a, 2) and (3,-6) is parallel to the line with slope 4.
 

5. Suppose lines y1 and y2 are perpendicular.

    a. What is the relationship between their slopes?

    b. Do you think this relationship always holds for perpendicular lines?
 

6. The following data was obtained experimentally by a research company. Determine if there is a linear relationship among all of the values. Explain your answer.
 
 

Hours watching TV per week

3

6

9

12

15

18

G.P.A.

3.8

3.5

3.2

2.9

2.5

2.0

7. If the horizontal distance from the edge of the roof on your house to the center of the roof is 24 feet and the roof has a 35% grade, by how much does your roof rise from its lowest to its highest point?
 
 

Congratulations! You've completed Part I of the first module.

 


 
 

PART II--THE SLOPE-INTERCEPT FORM OF A LINE
 

You will be working with the slope-intercept form of a line y = mx + b.
 
  Display the graph as necessary to answer the questions below.     

Graphs, such as this one, with the phrase 'ZOOM FEATURE ACTIVE'  at the bottom, are scalable.  See  zoom  for the directions for using this feature.
 

Critical Thinking Questions
 

1. Let m1 = 1.5 and b1 = 3.

    a. For every unit x changes, by how many units does y change?

    b. What is the slope of the line?

    c. What are the x and y intercepts? Plug in the x-value where the graph crosses the y-axis to find one intercept.Hint
 

2. Which part of the equation y = mx + b indicates the slope? Which part of the equation represents the y-intercept?
 

3. What is the equation of the line with slope - and y-intercept -7?
 

4. Now let m 2 = 2.5 and b2 = 3.

    a. What are the x and y intercepts of this line?

    b. If x changes by 3 units, by how many units does y change?
 

5. Now graph a third line with m3 = 2.5 and b3 = 7.5. What are the x- and y-intercepts of this line?
 

6. What is the x-intercept of the equation y = mx + b?
 

7. Now let y1 = 0.4x - 3 and y2 = x + 4.

    a. What is the relationship between the two lines?

    b. Why are they related this way?
 

8. Now let y1 = x and y2 = -x + 4.

    a. What is the relationship between the two lines?

    b. Why are they related this way?
 
 

Skill Exercises
 
 

1. Rewrite the linear equation 3x - 2y - 6 = 0 in slope-intercept form. State its slope and y-intercept.
 

2. For each of the following lines:

    a. Fill in the table below.
 

If any of the values do not exist, write DNE.
 
 

.

y = -2x + 6

x = 6

y = -4

2x - 3y = 0

Slope

x-intercept

y-intercept

    b. Graph each line, using the grids in your workbook.
 

3. Consider the lines 2x - 3y = 11 and 3x + 2y = 22 .

    a. How are they related?

    b. Why are they related this way?
 

4. Write the equation of the horizontal line passing through (2, 3).
 

5. Write the equation of the line which passes through (4, 2) and (0, -3).
 

6. A line has x-intercept 4 and y-intercept -3. Write the equation of the line in slope intercept form.
 

7. Water freezes at 0 Celsius and 32 Fahrenheit and boils at 100 C and 212 F.

    a. Derive a linear relationship between the Celsius and Fahrenheit temperature scales.

   Use the variables C and F.

    b. Which Celsius temperature corresponds to 4 below zero on the Fahrenheit scale?
 
 

 


 
 

PART III--THE POINT-SLOPE FORMULA
 

You will now work with the point-slope formula, y - y1 = m(x - x1) , where (x1, y1) is a known point.
 
Display the graph as necessary to help answer the questions below.     
 

Critical Thinking Questions
 

1. Given the line y - 4 = -2(x + 7) ,

    a. Which point on the line can be read directly from the equation?

    b. What is the slope of the line?

    c. Where does the line cross the x-axis?

    d. Where does the line cross the y-axis?
 

2. Identify the y-intercept of y - y1 = m(x - x1) .
 

3. Find the equation of the line passing through the point (-1, 4) which is perpendicular to the line x = 8 .
 

4. Find the equation of the line passing through the origin with slope 5.
 

5. Explain in words how you would use the point-slope formula to find the equation of a line passing through two points.First, use both points to find the ______.Hint
 

6. Find the equation of the line passing through the points (10, 18) and (-15, 20) and express your answer in the general form Ax + By +    C = 0.
 
 

Skill Exercises
 
 

1. Write the equation of the line passing through (-1,4) which has slope 3.
 

2. Find the equation of the line passing through (-2,5) and (6,-7).

   Express your answer in slope-intercept form.
 

3. If a line with slope 1 passes through (-2,0), what is the y-intercept ?
 

4. Given the line 3x + 2y - 6 = 0

    a. What are its slope and y-intercept?

    Show your work.

    b. Does the point (-2,6) lie on the line?
 

5. What is the equation of the line perpendicular to the line in Q.4 above which passes through the point (-2,3)? Perpendicular lines have slopes that are _______ _______ of each other.