Math 2 Honors
Lesson 2-3: Patterns in Tables, Graphs, and Rules
Name ________________________________
Date _____________________________
Learning Goals:
 I can identify a quadratic expression, ax2+bx+c.
 I can predict whether a quadratic will have a minimum or a maximum based on the value of a.
 I can identify a parabola’s reflective symmetry in a table or graph.
 I can explain why f(x) + k translates the original graph of f(x) up k units and why f(x) − k translates
the original graph of f(x) down k units.
 I can explain why k ∙ f(x) vertically stretches or compresses the graph of f(x) by a factor of k and
predict whether a given value of k will cause a stretch or a compression.
In questions 1 – 3, you will be exploring the equation f ( x )  ax 2 , when a is positive.
1. Complete the following tables for the function rules, and then graph both rules on the coordinate
plane. Use 2 different colors for the functions.
g ( x)  x 2
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
g(x)
h( x )  2 x 2
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
h(x)
a. What do the rules have in common? How are they different?
b. What do the graphs have in common? How are they different?
c. What do the tables have in common? How are they different?
d. Graph additional parabolas in your calculator by experimenting with a-values that are larger
than 2. Generalize what happens to the shape of the parabola as the value of “a” increases.
OVER 
Page 2
2. Use the following rules, tables and graph to answer parts a – c.
a. What do the rules have in common? How are they different?
b. What do the graphs have in common? How are they different?
c. What do the tables have in common? How are they different?
d. Graph additional parabolas in your calculator by experimenting with a-values that are between 0
and 1. Generalize what happens to the shape of the parabola as the value of “a” gets closer to
zero.
3. Use the following rules, tables and graph to answer parts a – c.
r ( x)   x 2
s ( x)   2 x 2
a.
What do the rules have in common? How are they different?
b.
What do the graphs have in common? How are they different?
c.
What do the tables have in common? How are they different?
Page 3
In question 4, you will be exploring the equation, f ( x)  ax  c. when a is positive.
2
4. The following graphs are in the form f ( x )  ax 2  c.
I
II
I
II
Match the graph with its equation.
Choices:
Equations
Graph I
___________________
f ( x)  x2  7
Graph II
___________________
f ( x)  x2  7
Graph III ___________________
f ( x)  x2  3
III
f ( x)  x2  3
f ( x)  x2
f ( x)  x
f ( x)  x  7
f ( x)  x  3
a. In complete sentences, explain your reasoning for your choices when you matched the equations
to the graphs above.
b. How did a positive c value affect the graph?
c. How did a negative c value affect the graph?
d. Given the function b( x )  3x 2 , write the function rule that shifts the graph 5 units up.
e. Given the function m( x )  4 x 2  3 , write the function rule that shifts the graph 6 units down.
OVER 
Page 4
Summarize the Mathematics
In this investigation, you discovered some facts about the ways that patterns in tables and graphs of
quadratic functions f ( x)  ax2  bx  c  a  0 are determined by the values of a and c.
a. What does the sign of “a” tell about the patterns of change and graphs of quadratic functions given
by rules in the form f ( x)  ax 2 ?
What does the absolute value (“number part”) of “a” tell you?
b. How are the patterns of change and graphs of quadratic functions given by rules like f ( x)  ax 2  c
related to those of the basic quadratic function f ( x)  ax 2 ?
What does the value of “c” tell about the graph?
NOTES:
Check Your Understanding
Use what you know about the relationship between rules and graphs for quadratic functions to match the
graph with the correct function.
Rule I
Rule IV
f ( x)  x 2  2
f ( x)  x 2  2
Rule ________
Explanation:
Rule II
Rule V
f ( x)   x 2  2
f ( x)  0.5x 2  2
Rule ________
Explanation:
Rule III
Rule VI
f ( x)  0.5x 2  2
f ( x)  15 x 2  2
Rule ________
Explanation: