Discovering the Relationship Between a Function and its Graph
Part I: Exploration
Let’s look at several functions and their graphs. What do you notice about the sets of
functions and graphs that match and the sets that do not?
This graph and equation match:
y  2x  3

This graph and equation do not match:
y  5x 1

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Discovering the Relationship Between a Function and its Graph
This graph and equation match:
1
y  x 2
2

This graph and equation do not match:
5
y  3x 
2

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Discovering the Relationship Between a Function and its Graph
This graph and equation match:
1
y  x3
3

This graph and equation do not match:
y  3x  4

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Discovering the Relationship Between a Function and its Graph
This graph and equation match:
y  4x

This graph and equation do not match:
1
2
y  x
3
5

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Discovering the Relationship Between a Function and its Graph
This graph and equation match:
y  x 2

This graph and equation do not match:
3
y  x 1
2

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Discovering the Relationship Between a Function and its Graph
This graph and equation match
y  1 2 x

This graph and equation do not match
y  1 3x

This graph and equation match
y  1 (.5) x

This graph and equation do not match
y  3 2 x

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Discovering the Relationship Between a Function and its Graph
This graph and equation match
y  3 3x
This graph and equation do not match
2 x
y  5  
5 


This graph and equation match
y  4 (.25) x

This graph and equation do not match
y  4 (.6) x

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Discovering the Relationship Between a Function and its Graph
This graph and equation match
 3 
y    4 x
 4 

This graph and equation do not match
y  (.5) (.85) x

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Discovering the Relationship Between a Function and its Graph
Part 2: Linear Functions
1. Make at least 3 observations about the graphs and equations that match.
a.
b.
c.
2. Make at least 3 observations about the graphs and equations that do not match.
a.
b.
c.
3. Write a rule that will help other students determine whether or not a graph
matches an equation:
4. The general form of a linear function is y  mx  b. From you observations,
a. What does the m represent? How does the m value affect the graph?

b. What does the b represent? How does the b value affect the graph?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Discovering the Relationship Between a Function and its Graph
Part 3: Exponential Functions
1. Make at least 3 observations about the graphs and equations that match.
a.
b.
c.
2. Make at least 3 observations about the graphs and equations that do not match.
a.
b.
c.
3. The general form of a linear function is y  a b x . From you observations,
a. What does the a represent? How does the a value affect the graph?

b. What does the b represent? How does the b value affect the graph?
4. Complete the following table:
Sign of a
+
Value of b
>1
+
0<b<1
-
>1
-
0<b<1
Appearance of Graph
Positive y-intercept, y-values increase
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.