What you should learn

To graph a line given any linear equation

NCTM Curriculm Standards 2, 4, 6 - 10

 

Introduction: Solome was flying from New York to Spain to be an exchange student for the spring quarter. After climbing to cruising level, the pilot announced that they were at an altitude of 13,700 meters and the outside temperature was about -76 degrees Celsius. You may think that since you get closer to the sun as you get farther from Earth, you might get warmer as altitude increases. However, as altitude increases, the air gets thinner and colder.

The air temperature above Earth on a day when the temperature at ground level is 15 degrees Celsius can be calculated using the formula a + 150t = 2250, where a is the altitude in meters and t is the temperature in degrees Celsius.

Suppose you wanted to graph a + 150t = 2250. In Chapter 5, you learned that you could graph a linear equation by finding and then graphing several ordered pairs that satisfy the equation.

Using the x and y intercepts is a convenient way to find ordered pairs to graph a linear equation. In a + 150t = 2250, these can be called the a-intercept and the t-intercept.

Find the a-intercept

a + 150t = 2250

a + 150(0) = 2250

a = 2250

Find the t-intercept

a + 150t = 2250

(0) + 150t = 2250

150t = 2250

t = 15

Now graph (2250, 0) and (0, 15), the ordered pairs fo rhte intercepts. Draw the line representing the equation by connecting the points. Thus, you can graph a line if you know two points on that line.

Use the equation to confirm the pilot's announcement that the temperature was -76 degrees Celsius at 13,700 meters altitude.

a + 150t = 2250

13,700 + 150t = 2250

150t = -11,450

t = -76.33

So the pilor was correct in saying that the outside temperature was about -76 degrees Celsius.

You can also graph a line if you know its slope and a point on the line. The forms of linear equations you have learned in this chapter can help you find the slope and a point on the line.

Exercise 1: Graph y - 1 = 3(x + 2)

This equation is in point-slope form. The slope is 3, and a point on the line is (-2, 1).

Graph the point (-2, 1). Remember that the slope represents the change in y and x.

Thus, from the point (-2, 1) , you can move up 3 and right 1. Draw another dot. You can repeat this process to find another point on the graph. Draw a line connecting the points. This line is the graph of y - 1 = 3(x + 2).

Check: Check (-1, 4) to make sure that it satisfies the equation.

y - 1 = 3(x + 2)

4 - 1 = 3(-1 + 2)

3 = 3

 

 

You can also use the slope-intercept form to help you graph an equation.

Exercise 2: Graph (3/4)x + (1/2)y = 4.

You frequently need to rewrite the equations before you can determine the best way to graph them.

Exercise 3: Graph (4/5)(2x - y) = 6x + (2/5)y - 10

It is often helpful to make a sketch of the graph of an equation when you are trying to observe patterns and make predictions.

Exercise 4: The mule deer likes to feed upon the bitterbrush plant, which is native to the dry regions of western North America. The diameter of each bitterbrush twig is related to the length of teh twig by the function l = 1.25 + 8.983d, where l is the length(inches) if the twig, d is the diameter in inches, and d 0.1

a. Graph the function. Name the independent and dependent variables.

b. Describe the relationship between a twig's diameter and length.

c. What would be the length of a twig if its diameter is 0.5 inch?

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: 15 - 35 odd, 36, 37, 39 - 45

Alternative Homework: Enriched: 14 - 36 even, 37 - 45

Extra Practice: Students book page 770 Lesson 6-5

Extra Practice Worksheet: Click Here.

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