Course: 8th Grade Math DETAIL LESSON PLAN Student Objective 8.F.A.3-1 Interpret the equation y=mx + b as defining a linear function, whose graph is a straight line 8.F.A.3-2 Give examples of functions that are not linear, meaning that the points when graphed do not form a straight line.

Lesson Identifying Linear and Non-Linear Functions Homework Teacher selected

Bellwork Teacher selected Prior Knowledge

Review bellwork

Review homework Introduction TODAY, we will learn how to identify and graph linear and non-linear equations.

Teacher Input

Pass out student notes.

Talk a little bit about the real world example of “linear”.

Review student notes on linear equations.

Talk a little bit about the real world example of “non-linear”.

Review student notes on non-linear equations.

Classwork: Linear and Non-linear Worksheet

Assessment Question the students for understanding. Observe students as they work on classwork/homework.

Closure Teacher selected

Linear equations are equations that make a straight line when graphed!

Nonlinear equations are equations that DO NOT make a straight line when graphed.

U-Shape V-Shape S-Shape

For the adventurous “Zip Lining” can be a great way to do a

little sightseeing. The steepness of the zip line is the

slope. Slope is an example of a relationship that is linear! Hawaii

Linear equations are equations that make a straight line when graphed.

Slope-Intercept Format

Any equation in what we call “Slope-Intercept” form will always graph to be a straight-line.

Slope-Intercept Form

slope y-intercept slope y-intercept

Examples – Below are two equations in slope intercept form. Complete the function tables below, and then graph. What shape do the graphs make?

1) 2)

Linear Equations can be presented in forms other than y = mx + b. Examples: Complete the function table, and then graph. What shape do the graphs make?

1)

Question: Could this equation be changed so that it is written in slope-intercept form?

2)

Question: How about this equation, could it be changed so that it is in slope-intercept form?

When you kick a ball in the air, the path that the ball

follows is a curve called a parabola.

A parabola is an example of a relationship that is not linear.

Nonlinear equations are equations that DO NOT make a straight line when graphed.

Today, we will discuss three types of non-linear equations that graph to form the shape of a “U”, “V”, or “S”.

Graphing Non-Linear Equations Let’s take a look…

Complete each table, and then graph. What shape does the graph make?

1) Shape_____

2) Shape_____

3) Shape_____

Characteristics of Non-Linear Shapes U-shaped If the x is squared, x², the equation when graphed will form a U-shaped curve called a parabola.

Rhyme: The power of 2 will make a U.

y = x² + 5 If the x is positive the U will open up.

y = -x² + 2 If the x is negative the U will open downward.

V-shaped If the x is an absolute value, |x|, the equation when graphed will form a V-shaped figure.

Trick: The symbol | | snaps to form a V.

y = |x| + 5 If there is a positive outside the || the V will open up.

y = -|x| + 2 If there is a negative outside the || the V will open downward.

S-shaped

If the x is cubed, x³, the equation when graphed will form an S-shaped figure.

These equations will form an S-shape. y = x³ + 2, 2 x³+1

Example graphs of an S-shape.

Name: _______________________ Linear & Non-Linear Classwork Period: _________ Equations

Directions: Without graphing, predict the shape of the following equations. (straight line, u-shaped, v-shaped, or s-shaped)

1. y = |x| + 1 __________________ 11. y x 6 __________________

2. y = 3x + 4 __________________ 12. 6x + 3y = 12 __________________

3. y x 8 __________________ 13. y 2x 6 __________________

4. y 3 x __________________ 14. 4x + y = 1 __________________

5. y = x³ __________________ 15. y 12x + 7 __________________

6. y x 3 __________________ 16. y x³ + 4 __________________

7. y 2x 2 __________________ 17. y x __________________

8. y 5x 3 __________________ 18. y 4 2x __________________

9. y =

x 2 __________________ 19. y = 3 + x² __________________

10. y 2x³ +1 __________________ 20. y =

x² + 1 __________________

21. Describe each equation as linear or non-linear.

a. y = 8 × 5x __________________

b. y = (3x + 10)² __________________

c. y = 11x² __________________

d. y = 12x – 3 __________________

e. y = 3x³ + 2 __________________

Directions: For each function, complete the table for integer values of x from -2 to 2. Then graph each function and name the shape. Name Shape

1)

2)

3)

4)

Answer Key

Linear equations are equations that make a straight line when graphed.

Slope-Intercept Format

Any equation in what we call “Slope-Intercept” form will always graph to be a straight-line.

Slope-Intercept Form

slope y-intercept slope y-intercept

Examples – Below are two equations in slope intercept form. Complete the function tables below, and then graph. What shape do the graphs make?

1) 2)

Linear Equations can be presented in forms other than y = mx + b. Examples: Complete the function table, and then graph. What shape do the graphs make?

1)

Question: Could this equation be changed so that it is written in slope-intercept form?

Yes

. So, the point is that any equation that can be

rewritten in slope- intercept form will graph to be a straight line.

2)

Question: How about this equation, could it be changed so that it is in slope-intercept form?

Yes Here again, any equation that can be rewritten in slope- intercept form will graph to be a straight Line.

Nonlinear equations are equations that DO NOT make a straight line when graphed.

Today, we will discuss three types of non-linear equations that graph to form the shape of a “U”, “V”, or “S”.

Graphing Non-Linear Equations Let’s take a look…

Complete each table, and then graph. What shape does the graph make?

1) Shape

2) Shape

3) Shape_____

Characteristics of Non-Linear Shapes U-shaped If the x is squared, x², the equation when graphed will form a U-shaped curve called a parabola.

Rhyme: The power of 2 will make a U.

y = x² + 5 If the x is positive the U will open up.

y = -x² + 2 If the x is negative the U will open downward.

V-shaped If the x is an absolute value, |x|, the equation when graphed will form a V-shaped figure.

Trick: The symbol | | snaps to form a V.

y = |x| + 5 If there is a positive outside the || the V will open up.

y = -|x| + 2 If there is a negative outside the || the V will open downward.

S-shaped

If the x is cubed, x³, the equation when graphed will form an S-shaped figure.

These equations will form an S-shape. y = x³ + 2, 2 x³+1

Example graphs of an S-shape.