# Algebra Introduction to Quadratic Functions December Mr Normile P Introduction

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```					Algebra 1B                               December 15, 2008                               Mr. Normile
Introduction to Quadratic Functions                                                             P. 1

Introduction to Quadratic Functions
So far, this course has dealt mostly with linear functions. Today’s assignment introduces a
different type of function called a quadratic function. Briefly here are the main things to know
about quadratic functions:
•   The function formula will always contain an x2.
•   The graph will always be a curve instead of a straight line.
When you reach page 4 you’ll get some additional vocabulary and facts about quadratic functions.

Making tables and graphs of quadratic functions
1. a. Without a calculator, make an input-output table and a graph for the function f(x) = x2.
For your inputs, include some negative numbers, zero, and some positive numbers.
After you’ve drawn several points, connect them with a curved graph.
x           f(x)

b. Make the table and graph of f(x) = x2 again, using your calculator. Record your work by
filling in the screen pictures below.
Algebra 1B                              December 15, 2008                              Mr. Normile
Introduction to Quadratic Functions                                                           P. 2

2. a. Without a calculator, make an input-output table and a graph for the function f(x) = x2 – 5.
For your inputs, include some negative numbers, zero, and some positive numbers.
x           f(x)

b. Make the table and graph of f(x) = x2 – 5 again, using your calculator. Record your work
by filling in the screen pictures below.
Algebra 1B                              December 15, 2008                              Mr. Normile
Introduction to Quadratic Functions                                                           P. 3

3. a. Without a calculator, make an input-output table and a graph for the function f(x) = –2 · x2.
For your inputs, include some negative numbers, zero, and some positive numbers.
x           f(x)

b. Make the table and graph of f(x) = –2 · x2 again, using your calculator. Record your work
by filling in the screen pictures below.

4. In problems 1 through 3, you graphed three quadratic functions. Based on what you’ve seen
so far, what kinds of shapes do quadratic function graphs have? Use your own words and
pictures in your answer.
Algebra 1B                               December 15, 2008                                Mr. Normile
Introduction to Quadratic Functions                                                              P. 4

Key vocabulary and facts
•   A quadratic function is any function whose formula has the form f(x) = ax2 + bx + c.
(The bx and c terms are optional; that is, b or c could just be 0.)
•   The graph of a quadratic function is always shaped like ∪ or ∩, and called a parabola.
•   The shape will be ∪ if a is positive. The shape will be ∩ if a is negative.
•   The lowest point of the ∪ or the highest point of the ∩ is called the vertex.

Another table and graph
5. Answer these questions about the function f(x) = 1x2 – 2x.
a. Just from looking at the numbers in the function formula, which of the parabola shapes
(∪ or ∩) will the graph of this function have? Tell how you know.

b. Without a calculator, make an input-output table and a graph for the function.
x                f(x)

c. Find the vertex (highest or lowest point). Circle that point on both the graph and the table.
Algebra 1B                              December 15, 2008                              Mr. Normile
Introduction to Quadratic Functions                                                           P. 5

Application problem
Many things in everyday life and in nature are described by quadratic functions and have
parabola shapes, such as in the following problem.
6. Kim throws a ball out a second-floor window. It flies through the air then lands on the
ground. Let x = the time since she threw the ball, in seconds.
Let f(x) = the height of the ball above the ground, in feet.
The function formula is f(x) = –16x2 + 24x + 16.
a. Evaluate f(0). What is the meaning of the answer in the problem situation?

b. Complete the input-output table given below (you may use your calculator to help;
for example, you could ask it to calculate -16*0.252 + 24*0.25 + 16 ).
Then make a graph (the grid shown goes from 0 to 5 on the x-axis, 0 to 30 on the y-axis).
x                f(x)

0
0.25
0.5
0.75
1
1.25
1.5
1.75
2

c. What was the highest height reached by the ball?

d. For how much time did the ball stay in the air?
Algebra 1B                              December 15, 2008                               Mr. Normile
Introduction to Quadratic Functions                                                            P. 6

More tables and graphs
7. Answer these questions about the quadratic function f(x) = –2x2 + 8.
a. Which of the parabola shapes (∪ or ∩) will the graph of this function have? Tell why.

b. Make an input-output table and a graph. Use your calculator as little as possible. (Hint:
back in October we learned something about squaring a negative number with a
calculator.)
x                f(x)

c. What are the coordinates of the vertex?

d. How many x-intercepts does this graph have? What are the coordinates of the x-intercept
point(s)?

e. How many y-intercepts does this graph have? What are the coordinates of the y-intercept
point(s)?
Algebra 1B                              December 15, 2008                               Mr. Normile
Introduction to Quadratic Functions                                                            P. 7

8. Identify the shapes of these function graphs without actually making the graphs. For each
function, put a check mark in one of the four columns. (If a function is neither linear nor
quadratic, check “none of these.”)
Hints: Remember x2 is the same as 1x2, –x2 is the same as –1x2.
straight line     ∪ parabola     ∩ parabola         none of these
f(x) = x2 [example]                               

f(x) = –2x + 4

f(x) = –3x2 + 4x – 5

f(x) = –x2 + 6x + 7

f(x) = x2 – 2x

f(x) = 2x

f(x) = 8 – x

f(x) = 8 – x2

9. Graph these quadratic functions on your calculator. You will sometimes need to use
WINDOW to adjust the screen so that the ∪ or ∩ shape, including the vertex, can be seen on
the screen. Record window settings, and sketch each graph as you see it on the screen. If
you’re having trouble figuring out what window settings to use, press  to see what kinds
of numbers are in the table.
a. f(x) = x2 – 22x + 121
WINDOW
Xmin =
Xmax =
Ymin =
Ymax =
b. f(x) = x2 + 20
WINDOW
Xmin =
Xmax =
Ymin =
Ymax =
Algebra 1B                            December 15, 2008   Mr. Normile
Introduction to Quadratic Functions                              P. 8

c. f(x) = –x2 + x – 15
WINDOW
Xmin =
Xmax =
Ymin =
Ymax =
d. f(x) = x2                 WINDOW
Xmin =
Xmax =
Ymin =
Ymax =
e. f(x) = –2x2 + 12x
WINDOW
Xmin =
Xmax =
Ymin =
Ymax =

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