IMP 3 Function Activity
Linear, Quadratic and Cubic Functions
with some extension to higher degree polynomials: by Ms. Wiseman
Directions:
1) Graphing:
 Use a graphing calculator to do all the graphing. This will save you precious time and
energy!
 Graph the parent function in y1.
 Graph the transformed function one at a time in y2. This will help you “see” the changes
in the functions.

Graph the transformed functions in order. You do not have to draw any of your graphs. However, you
may wish to sketch a few for future use.

Look for patterns (similarities/differences in and between sets) that will help you answer
the questions.*
2) Completing the POW:
 Your results/answers to any italicized questions or directions must be written in your
POW.
 The results should be in the order of the directions/questions.
 The results/answers should be written in complete sentences.
 Any tables may be completed on this sheet.
3
* If you already know the patterns or figure out the pattern before you have graphed all the
transformed functions, you may go directly to answering the questions.
Due Date: 1-1-2358
Linear functions:
Parent function: y = x
Transformed function: y = ax + b (often learned as y = mx + b)
Set A:
y=2x
y = 5x
y = 10x
y = .5 x
y = .05 x
Set B:
y = -2x
y = -5x
y = - 10 x
y = -.5x
Set C:
y=x+2
y=x+5
y = x + 10
Set D:
y=x-2
y=x-5
y = x - 10
1. If a is positive will the graph be increasing or decreasing? Why?
2. If a is negative, will the graph be increasing or decreasing? Why?
3. If |a| >1 (If a is a large negative OR positive number), how will the graph change?
4. If |a| < 1 (If a is a small negative OR positive number), how will the graph change?
5. If b is positive, how is the graph transformed?
6. If b is negative, how will the graph be transformed?
7. Describe how you would recognize a linear graph.
8. Complete the tables below:
y = 3x
y = 5x + 4
What is the value of a?
X
Y
-1
0
1
2
What is the a-value?
X
Y
-1
0
1
2
9. How does the a-value relate to the difference between the y-values (as x increases by 1)?
10. Why does this relationship define a linear function?
11. How do we refer to the a-value when we discuss linear functions? What word do we use?
12. What is the formula for finding this relationship?
13. If the x-values in a table did NOT increase by 1, how could we tell if a function was linear or
not? Use your procedure to show that the table below represents a linear function.
X
5
13
19
21
28
Y
31
71
101
111
146
14.
15.
16.
17.
How would you recognize a linear relationship in a word problem?
How would you identify the b-value from a table when x=0 is given?
How could you calculate the b-value algebraically when x = 0 is not given in the table?
What is another term for the b-value?
******************************************************************************
Quadratic Functions:
Part 1 Parent function: y = x2
Transformed standard form function: y = a x2 + bx + c
Set A
y = 2x2
y = -x2
y = -2x2
y = 10x2
y = 0.5x2
y = 0.05x2
Set B
y = x2 + 2x
y = x2 + 5x
y = x2 - 2x
y = x2 - 5x
y = x2 + 0.5x
Set C
y = x2 + 2
y = x2 + 5
y = x2 - 2
y = x2 - 5
Set D
y = x2 + 2x + 3
y = x2 + 5x + 3
y = x2 + 2x - 3
y = x2 + 5 x -3
y = x2 + 2x + 5
y = x2 + 2x -5
y = x2 - 2x + 5
1. If a is positive, what is true about the shape of the transformed graph?
2. If a is negative, what is true about the shape of the transformed graph?
3. If |a| > 1, what is true about the shape of the transformed graph?
4. If |a| < 1, what is true about the shape of the transformed graph?
5. If c is positive, how is the graph transformed?
6. If c is negative, how is the graph transformed?
HONORS:
7. How do a and b interact to transform the graph? Be detailed.
Part 2 –
Parent function: y = x2
Transformed vertex form function: y = a(x – h)2 + k
Set A
Y = (x – 5)2
Y = (x + 10)2
Y = (x – 3)2
Set B
y = x2 – 5
y = x2 + 3
y = x2 + 6
Set C
y = (x – 5)2 – 5
y = (x + 10)2 + 3
y = (x-3)3 + 6
Set D
y = 4(x – 5)2
y = 2(x – 5)2
y = ½ (x – 5)2
Y = -3(x – 5)2
8. How do the h and k values affect the graph of y = x2? Be specific, discussing positive and
negative values.
9. How does the a-value affect the graph? If you can reference a previous answer, please feel
free to do so. (i.e.: “See answer #75”)
Part 3 10. Complete the tables for each function below. Then, find the first and second differences of
the y-values.
y = x2
y = 3x2 + 4
X
-2
-1
0
1
2
Y
X
-2
-1
0
1
2
Y
11. Describe how you would recognize a quadratic relationship in a word problem.
12. What is one of the most common geometric applications for quadratic functions?
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Cubic Functions:
Part 1 Parent Function: y = x3
Transformed standard form Function: y = ax3 + bx2 + cx + d
Set A
y = 2x3
y = 5x3
y = 0.5x3
y = -x3
y = -5x3
Set B
y = x3 + 2
y = x3 + 5
y = x3 – 4
y = x3 – 10
Set C
y = x 3 + x2
y = x3 – x2
y = x3 + 3x2
y = x3 – 3x2
y = x3 + 0.5x2
Set D
y = x3 + x
y = x3 – x
y = x3 + 3x
y = x3 – 3x
y = x3 + 5 x
1. If a is positive, what is true about the shape of the transformed graph?
2. If a is negative, what is true about the shape of the transformed graph?
3. If |a| > 1, what is true about the shape of the transformed graph?
4. If |a| < 1, what is true about the shape of the transformed graph?
5. If d is positive, how is the graph transformed?
6. If d is negative, how is the graph transformed?
HONORS:
7. Now, experiment with several functions in which a, b, c and d are all changed at the same
time. What do you notice? You might look carefully at the point of inflection.
Part 2 –
Parent Function: y = x3
Transformed function in a different form: y = a(x – h)3 + k
Set A
Set B
Set C
3
y = (x – 5)3
y=x –5
y = (x – 5)3 – 5
y= (x + 10)3
y = x3 + 3
y = (x + 10)3 + 3
y= (x – 3)3
y = x3 + 6
y = (x-3)3 + 6
Set D
y = 4(x – 5)3
y = 2(x – 5)3
y = ½ (x – 5)3
y = -3(x – 5)3
8. How do the h and k values affect the graph of y = x2? Be specific, discussing positive and
negative values. You might look carefully at the point of inflection.
9. How does the a-value affect the graph? If you can reference a previous answer, please feel
free to do so. (i.e.: “See answer #75”)
Part 3 10. Complete the x-y table below for at least two of the functions above.
11. Find the first, second and third differences of the y-values.
12. How would the third differences help you identify a cubic function?
X
Y
-2
-1
0
1
2
13. What common geometric application uses cubic functions?
14. How could we use our calculators to find the x-values where the graph of a cubic crosses the
x-axis?
15. How could we extend the zero product rule to find the zeros of the equation below? Please
state the zeros.
0 =(3x + 18)(x – 9)(2x – 16)
Extension for everyone:
16. How can you calculate the y-intercept of any function?
HONORS
The degree of a function is the value of its largest exponent.
17. Using your results from the past three explorations, how could you determine if you have a
5th degree function?
18. An nth degree function?
19. How could we extend the zero product rule to a 5th degree function? A 6th degree function?
an nth degree function?
20. End Behavior of Polynomials
a. Graph the following functions:
y = x4
y = -x4
y = x5
y = -x5
6
6
7
y=x
y = -x
y=x
y = -x7
y = 0.5x4 + x3 – 3x + 2
y = -2x5 – x4 + 5x3 + x2 - 3
(Graph more functions with higher degrees if you need more examples.)
b. Based on your work throughout this POW, complete the table below by cutting and pasting
(or re-drawing) the four possible polynomial function shapes. The (……) mean that we are
concerned only with the ends of the shape, not the “wiggle” in the middle.
Leading Coefficient is Positive
Leading coefficient is Negative
Degree
is odd
Degree
is even
End Behavior Shapes:
A.
B.
………..
………..
C.
D.
………..
………..
Vocabulary terms:
Point of inflection – The point on any graph where the curve changes from concave up to
concave down.
Concave up:
Concave down
Point of inflection
First, second and third differences of the y-values – The first differences are the differences
between the actual y-values (when x increases by one).
The second differences are the differences between the differences. Etc.
Example: (The parentheses show the calculation!)
X
0
Y
1
1
4
2
16
1st Difference
2nd Difference 3rd Difference
3 (4-1)
9 (12-3)
12 (16-4)
27 (36 – 9)
36
48
3
64
4
256
5
1024
108
144
192
432
576
768
Increasing function - If a function is Increasing, the y-values get bigger as the x- values get
bigger.
The graph goes up from left to right.
Decreasing function - If a function is decreasing, the y-values get smaller as the x-values get
bigger. The graph goes down from left to right.
End Behavior – The shape (or behavior) of a graph/function when the x-values are really big
positive numbers or really big negative numbers.