The Shapes of Algebra

To find and use rules that relate independent and dependent variables or that predict change in one variable over time, it helps to be familiar with the table and graph patterns associated with various symbolic forms. As you work on the explorations of this investigation, look for answers to this question:

How do the forms of algebraic rules give useful information about the patterns in tables and graphs produced by these rules?

For each of the following explorations, consider the following suggestions:

1. For each rule in the set, produce a table of (x, y) values with integer values of x and graphs of (x, y) values for x between -5 and 5.

2. Record the table patterns and sketches of the graphs in your notes.

3. Compare the tables, graphs, and symbolic rules in the exploration. Note similarities, differences, and connections between the symbolic rules and the table and graph patterns. Explore some other similar rules to test your ideas.

4. Try to explain why the observed connection between rules and table/graph patterns makes sense.

Exploration #1: Compare the patterns of change in tables and graphs for these rules.

a) y=2x−4 b) y=−0 .5 x+2 c) y=0. 5 x+2 d)y=2x−4

Exploration #2: Compare the patterns of change in tables and graphs for these rules.a) y=|x| b) y=x+1 c) y=|x+1| d) y=−|x|+1

Exploration #3: Compare the patterns of change in tables and graphs for these rules.a) y=x2

b) y=2x c) y=−x2d) y=−x2+2

Exploration #4: Compare the patterns of change in tables and graphs for these rules.

a)y=1x b)

y= x3 c)

y=3x d)

y=−5x

Exploration #5: Compare the patterns of change in tables and graphs for these rules.a) y=3x b) y=x3

c) y=1 . 5x d) y=4x

Summarizing:

As a result of the explorations, you probably have some ideas about the patterns in tables of (x, y) values and the shapes of the graphs that can be expected for various symbolic rules. Summarize your conjectures in statements like these…

1) If we see a rule like …………, we expect to get a table like……..2) If we see a rule like …………., we expect to get a graph like…….3) If we see a graph pattern like ………., we expect to get a table like……….

Check Your Understanding:

Each item here gives three algebraic rules – one of which will have quite different table and graph patterns than the other two. In each case, spot the “alien” rule and explain how and why its graph and/or table pattern will look different from the other two.

a)y=10

x, y=10 x , y=x+10

b) y=x2+1, y=x+1 , y=1−x2

c) y=1 . 5x−4 , y=(1 .5x )−4 , y=2x d) y=1 . 5x−4 , y=0 .5 x−4 , y=−1. 5 x−4

Practice:1. Without the use of your graphing calculator, match the following six rule types to the

tables in parts a – e. Explain your reasoning in each case.

y=ax+b y = a|x|+b y=ax 2+b y=ax

y=ax

a)

b)

c)

d)

f)

x -4 -3 -2 -1 0 1 2 3 4 5y 18 11 6 3 2 3 6 11 18 27

x -4 -3 -2 -1 0 1 2 3 4 5y 16 14 12 10 8 6 4 2 0 -2

x -4 -3 -2 -1 0 1 2 3 4 5y 0.0625 0.125 0.25 0.5 1 2 4 8 16 32

x -4 -3 -2 -1 0 1 2 3 4 5y -

1.5-2 -3 -6 Error 6 3 2 1.5 1.2

x -4 -3 -2 -1 0 1 2 3 4 5y 13 10 7 4 1 4 7 10 13 16

2. Without the use of your graphing calculator, match the following six rule types to the graph sketches below. Explain your reasoning in each case.

y=ax+b y = a|x|+ b y=ax2+b y=ax

y=ax

a) b)

c) d)

e)

3. Without the use of your graphing calculator, sketch graphs you would expect to see from these rules. Explain your reasoning in each case.

a) y=7 x2+4 b) y=7−1

4x

c) y=4x−7 d) y= 4x e) y=4|x|+7

These activities are taken from Core-Plus Mathematics Copyright 2008.

Student Work:

Exploration #1

a)

b)

c)

d)

x y=2x−4

x y=−0.5x+2

x y=0.5x+2

x y=2x−4

Exploration #2

a)

b)

c)

d)

x y=|x|

x y=x+1

x y=|x+1|

x y=−|x|+1

Exploration #3

a)

b)

c)

d)

Exploration #4

x y=x2

x y=2x

x y=−x2

x y=−x2+2

a)

b)

c)

d)

x y=1x

x y= x3

x y=3x

x y=−5x

Exploration #5

a)

b)

c)

d)

x y=3x

x y=x3

x y=1.5x

x y=4x

Answer Key:

Exploration #1

a)

b)

c)

d)

x y=2x−4

-2 -8-1 -60 -41 -22 0

x y=−0.5x+2

-2 3-1 2.50 21 1.52 1

x y=0.5x+2

-2 1-1 1.50 21 2.52 3

x y=2x−4

-2 -5-1 -60 undefined1 -22 -3

Exploration #2

a)

b)

c)

d)

x y=|x|-2 2-1 10 01 12 2

x y=x+1

-2 -1-1 00 11 22 3

x y=|x+1|-2 1-1 00 11 22 3

x y=−|x|+1

-2 -1-1 00 11 02 -1

Exploration #3

a)

b)

c)

d)

Exploration #4

x y=x2

-2 4-1 10 01 12 4

x y=2x

-2 0.25-1 0.50 11 22 4

x y=−x2

-2 -4-1 -10 01 -12 -4

x y=−x2+2-2 -2-1 10 21 12 -2

a)

b)

c)

d)

x y=1x

-2 -0.5-1 -10 undefined1 12 0.5

x y= x3

-2 -2/3-1 -1/30 01 1/32 2/3

x y=3x

-2 -1.5-1 -30 undefined1 32 1.5

x y=−5x

-2 2.5-1 50 undefined1 -52 -2.5

Exploration #5

a)

b)

c)

d)

x y=3x

-2 1/9-1 1/30 11 32 9

x y=x3

-2 -8-1 -10 01 12 8

x y=1.5x

-2 4/9-1 2/30 11 1.52 2.25

x y=4x

-2 1/16-1 ¼0 11 42 16

Summarizing: Answers may vary

Check for understanding:

a)y=10

x, y=10 x , y=x+10

b) y=x2+1, y=x+1 , y=1−x2

c) y=1 . 5x−4 , y=(1 .5x )−4 , y=2x d) y=1 . 5x−4 , y=0 .5 x−4 , y=−1. 5 x−4

a) The two not circles will be a straight line while the alien will be undefined at 0 because the x is divided by 0.

b) The two not circled will be a U shape (one opens up and the other down) but the circled one will be a straight line because the x is not squared.

c) The two not circled will be an exponential growth while the other graph will be a straight line because the x is not in the exponent place.

d) While all three lines will be straight the two not circled will have a positive slope, while the other one is a negative slope.

Practice:

1)

y=ax+b b, because the table changes by the same amount each time

y=a|x|+b f, because the numbers change by the same amount on both sides of the minimum

y=a x2+b a, because the numbers are going to a minimum and increase by perfect squares

y=ax d, because there is an error at 0

y=ax c, because it goes up by increasing amounts

2)

y=ax+b a, because the graph is straight

y=a|x|+b d, because the graph makes a V shape

y=a x2+b c, because the graph makes a U shape

y=ax b, because there is no point at 0

y=ax c, because the graph goes up by increasing amounts

3)

a) parabola

b) linear

c) exponential growth

d) inverse

e) absolute values