What you should learn

To determine the x and y-intercepts of linear graphs from their equations

To write equations in slope-intercept form

To write and solve direct variation equations

NCTM Curriculm Standards 2, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

y-intercept

x-intercept

slope-intercept form

Introduction: Have you seen advertisements for banking in the newspaper? Some banks offer free checking. However in the fine print of the ad, you may find that free checking only occurs if you maintain a minimum daily balance of at least $2000.

Suppose that a bank like this charges a monthly service feet of $3 plus 10 cents for each check or withdrawal transcation if you fail to maintain a miniimum balance of $2000. The graph below shows the line taht represents this situation.

The coordinates at which a graph intersects the axes are known as the y-intercept and x-intercept. Since this graph intersects the y-axis at (0, 3), the y-intercept is 3. If you were to extend the graph to the left, you would find that it crosses the x-axis at (-30, 0), so the x-intercept is -30.

Exercise 1: Find the x and y intercepts of the graph of 3x + 4y = 6.

Remember that the x-intercept is the point at which y = 0

3x + 4(0) = 6

3x = 6

x = 2

Now let x = 0 to find the y-intercept.

3(0) + 4y = 6

4y = 6

y = 6/4 = 3/2

The x-intercept is 2, and the y-intercept is 3/2. This means that the graph crosses the x-axis at (2, 0) and the y-axis at (0, 3/2). you can use these points to graph the equation 3x + 4y = 6

 

Consider the graph below.

The line crosses the y-axis at (0, b), so its y-intercept is b. Write the point-slope form of an equation for this line using (o,b) as (x1, y1).

y - y1 = m(x - x1)

y - b = m(x - 0)

y - b = mx

y = mx + b

This form of an equation is called the slope-intercept form of a linear equation, because in this form the slope and y0intercept are easily identified.

Slope-Intercept Form of a Linear Equation: Given the slope m and the y-intercept b of a line, the slope-intercept form of an equation of the line is y = mx + b.

 

If you know the slope and y-intercept of a line, you can write an equationof the line in slope-intercept form. This equation can also be written in standard form.

Exercise 2: Write an equation of a line in slope-intercept form if the line has a slope of 2/3 and a y-intercept of 6. Then write the equation in standard form.

Let's compare the slope-intercept form with the standard form of a linear equation. First solve Ax + By = C for y.

Ax + By = C

Ax - Ax + By = C - Ax

By = C - Ax

y = (-Ax)/B + C/B

Thus, you can identify the slope and y-intercept from an equation written in standard form by using m = -A/B and b = C/B.

Exercise 3: Find the slope and y-intercept of the graph of 5x - 3y = 6

In Lesson 6-2, you learned how to use the point-slope form to find an equation of a line passing through two given points. Now you have another tool that you can use for the same situation.

Exercise 4: Write the slope-intercept and standard formsof the equation for a line that passes through (-3, -1) and (6, -4).

A special case ofthe slope-intercept form occurs when the y-intercept is 0. When m = k and b = 0, y = mx + b becomes y = kx. You may recognize this as the equation for direct variation. Another way to write this equation is y/x = k.

Exercise 5: Penny Dean holds the record for the fastest time swimming the English Channel. The 23-year-old Californian swam the 21-mile distance in 7 2/3 hours on July 29, 1978. At the same rate, how long would it take her to swim the Catalina Channel, a 26-mile distance near Los Angeles? (Use your problem solving tools).

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 19 - 53 odd, 54, 44, 57 - 67

Alternative Homework: Enriched: 20 - 52 even, 53 - 67

Extra Practice: Students book page 770 Lesson 6-4

Extra Practice Worksheet: Click Here.

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