Forms of Lines and Modeling Using Linear Equations

[Pages:44]UNIT 4 ? Math 621

Forms of Lines and Modeling Using Linear Equations

Description:

This unit focuses on different forms of linear equations. Slope- intercept, point-slope and standard forms are introduced. Students will write all three forms using short descriptions, tables of values and/or graphs. In addition, students will be able to transform one form into another and construct linear graphs for each form. They will be able to graph lines using the x and y- intercepts, a slope and y-intercept, a point and a slope or two points, accordingly to the given form of the equation.

The unit will be concluded with an introduction to linear modeling, where students will recognize linear relationship between independent and dependent variables. Students will derive equations, graph functions and use their equations and graphs to make predictions. In the process of modeling, students will determine the constant rate of change and initial value of a function from a description of a relationship. They will interpret slope as rate of change, relate slope to the steepness of a line, and learn that the sign of the slope indicates that a linear function is increasing if the slope is positive and decreasing if the slope is negative.

4.1 Slope and slope-intercept form of lines 4.2 Ivy Smith Performance Task

4.3 Point-Slope form of lines 4.4 Standard Form for lines Quiz on sections 4.1-4.4

4.5 Modeling using linear equations 4.6 More on Modeling 4.7 Unit Review

Unit 4 TEST

RESOURCES

4.1 Slope-intercept form of lines 4.2 Point-Slope form of lines 4.3 Standard Form for lines 4.4 Review of linear forms 4.5 Modeling using linear equations 4.6 More on Modeling

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4.1 Slope Formula and Slope-Intercept Form Review

The slope formula is:

m

=

y2 x2

- -

y1 x1

for the points (x1, y1) and (x2, y2 )

Example: Find the slope of the line that goes through the points (5, -3) and (2, 3)

1: Use the slope formula to find the slope of the line through the given points. Leave answers in simplified fractional form.

1. A(3, 1) and B(5, -1)

2. C(7, 0) and D(5, -3)

3. E(7, 8) and F(3, 8)

4. G(-3, 5) and H(-3, 2)

5. (2, 6) and (7, 1)

6. (4, -1) and (0, 9)

7. (3, 7) and (4, 7)

8. (7, 2) and (7, -4)

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9. (3, -3) and (2, -2)

10. (7, 0) and (2, 10)

Rewrite each of these equations in slope-intercept form.

11. 3x ? 4y = 24

12. x + 4y = 12

Use the slope formula to find the slope of the line going through each of these pairs of points.

13. A(7, 4) and B(-2, 1)

14. C(0, 3) and D(4, 0)

Identify the type of line (vertical, horizontal or oblique) for the given equations.

15. y = 3x + 4 16. 3x = 7y 17. 3y = 9 18. x = 3

vertical vertical vertical vertical

horizontal horizontal horizontal horizontal

oblique oblique oblique oblique

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Identify the slope and y-intercept of the lines represented by each of these equations.

19. y = 4x + 7

20. 3x + 2y = 12

slope = ________ y-intercept = ________ slope = ________ y-intercept = ________

21. x = 10

22. y = 3

slope = ________ y-intercept = ________ slope = ________ y-intercept = ________

Find the equation of each line in slope-intercept form 23. The line has a slope of 7 and a y-intercept at -2. ________________________________________ 24. The slope is -2 and it goes through the point (0, 3). ________________________________________ 25. The slope is 5 and it goes through the point (2, -1) ________________________________________

Write the equation of each line in slope-intercept form.

26.

27.

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Section 4.3 ? Point-Slope Form Notes and Practice Lines of the form _____________________ are written in ______________________ form. Example 1: Write an equation for the line with slope = 2, that goes through the point (7, 3)

Example 2: Write an equation for the line with slope = 2 , that goes through the point (5, -6) 3

Example 3: Write an equation for the line with slope = -4, that goes through the point (-2, 1)

In order to write the equation of any line you need _______________ and the ____________. If you are given both of those things, as you were above, then it's easy to use the point-slope form of the line. Sometimes you aren't actually given the information out-right, but you can figure it out from other information. Example 4: Write an equation for a line that goes through the points (1, 3) and (-2, 2). You have a point ? in fact, you have two points from which to choose, but you need __________________________________. Fortunately, you can compute it: Let's use the first point: __________________ with the slope ________________. The equation is: ___________________________________________________

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Example 5: Write an equation for a line that goes through the points (4, -5) and (2, 3): Example 6: Write an equation for a line that goes through the points (1, 3) and (7,3):

Example 7: Write an equation for a line that goes through the points (1, 3) and (1,4):

It is easy to graph lines that are written in point-slope form. Just plot the point and use the slope.

Example 8:

a. Graph the line y - 2 = 1 (x + 5) 2

b. Graph the line y - 3 = -2(x - 1)

The slope is: _______ a point is _________ The slope is: _________ a point is _________

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