Name: ____________________________________
Date: __________________
Introduction to Functions
Algebra 1
One of the most important concepts in all of mathematics is that of a function. The definition of a
function is given below.
Definition: A function is a rule, often expressed using an equation, graph, or table that gives
exactly one output for each input. The rule gives the value of a dependent variable given the value
of an independent variable.
Exercise #1: Roja was riding his bike from his house to the mall. The graph below gives Roja’s
distance from home, D, as a function of the time, t, since he left.
(a) State the independent (input) and the dependent
(output) variables.
t = 10 minutes
Distance (miles)
(b) Determine the distance that Roja is away from his
home at the following times:
8
6
4
2
D=
10
t = 35 minutes
D=
t = 45 minutes
D=
20
30
40
50
Time Since Leaving (minutes)
(c) For what values of t is Roja four miles away from home?
In this exercise, we see that Roja’s distance from home depends on how long he has been away from
home. Thus, his distance is a function of the time since he has departed.
Exercise #2: The equation y  2 x  5 gives the variable y as a function of the variable x. In this case
the value of y depends on the value of x.
(a) What is the output of this function when x has the following input values?
(i) x  4
(ii) x  6
(b) For what value of x is the output of this function y  17 ?
Algebra 1, Unit #1 – Algebraic Foundations – L10
The Arlington Algebra Project LaGrangeville, NY 12540
(iii) x  0
(iv) x  3
60
Exercise #3: A ball is dropped from the top of a 1000-foot high building. Its height, in feet, as a
function of time, in seconds since it was dropped, is given by the following equation:
h  1000  16t 2
(a) Use your calculator to produce a table of values for
this function. Then, plot these values on the axes
given.
1000
900
800
(b) Between what two consecutive integer times does the
ball fall to a height of 700 feet?
(c) Between what two consecutive integer times does the
ball hit the ground?
Ball Height (feet)
700
600
500
400
300
200
(d) By refining your calculator table, determine to the
nearest tenth of a second when the ball falls to a height
of 800 feet.
100
2
4
6
8
10
Time (seconds)
Exercise #4: Two functions have been entered into the graphing calculator. A calculator table for the
two functions is shown below for all integer values of x on the interval 2  x  4 .
(a) What is the output of the two functions when x = 3?
Y1 
Y2 
(b) For what value(s) of x is Y2  0 ?
(c) If these two functions were graphed on coordinate axes, at what coordinate points would they
intersect?
Algebra 1, Unit #1 – Algebraic Foundations – L10
The Arlington Algebra Project, LaGrangeville, NY 12540
Name: ____________________________________
Date: __________________
Introduction to Functions
Algebra 1 Homework
Skills
1. Determine the outputs for the function y  3x  7 given the following inputs.
(a) x  5
(b) x  0
(c) x  3
(b) x  10
2. Determine the outputs for the function y  x 2  4 given the following inputs.
(a) x  0
(b) x  3
(c) x  2
(d) x  2
3. Determine the outputs for the function y   x 2  2 x  3 given the following inputs.
(a) x  2
(b) x  3
(c) x  0
(d) x  2
4. The following calculator screen shot shows a table for a given function over the interval
4  x  2 . Answer the following questions based on this table.
(a) What is the output when the input is x  1 ?
(b) What is the output when the input is x  2 ?
(c) For what input value(s) will the output equal -1?
(d) What is the smallest (minimum) value that the function reaches over the interval 4  x  2 ?
Algebra 1, Unit #1 – Algebraic Foundations – L10
The Arlington Algebra Project LaGrangeville, NY 12540
Applications
5. A ball is thrown off the top of a building such that its height is given as a function of time since it
was thrown.
h (ft)
(a) What are the independent and dependent variables
120
for this function?
110
100
90
80
70
60
50
40
30
20
10
(b) At what height is the ball thrown from the roof?
(c) What is the value of t at which the ball reaches its
maximum height?
t (sec)
0
(d) What is the maximum height of the ball?
2
4
6
8
10
12
(e) What is the value of t at which the ball hits the ground?
6. The temperature of lake water over a 12 hour period is shown in the graph below. The temperature
is a function of the time since it was first measured.
(a) What is the minimum value of the function
over the interval 0  t  12 hours ?
Temp (F)
60
58
56
(b) What is the highest temperature the lake reaches?
54
52
(c) At what value(s) of t does the lake reach a
temperature of 54 F?
50
t (hrs)
0
Algebra 1, Unit #1 – Algebraic Foundations – L10
The Arlington Algebra Project, LaGrangeville, NY 12540
2
4
6
8
10
12