Sample Test Questions
(Barnett, Ziegler, and Byleen)
Chapter 3
Graphs and Functions
3-1
basic tools: circles
1.
What is the distance between the following sets of points?
2.
a.
(1,2) and (−3,4)
(5)
b.
(−3,4) and (4,−3)
(5)
c.
(−3,−2) and (−3,4)
(5)
Does the triangle defined by the three points (1,2), (5,1), and (6,3) form a right triangle?
Explain.
(10)
3.
What is the equation of a circle with a center at (3,4) and a radius of 3?
(5)
4.
For the circle given by:
(x−2)2 + (y+5)2 = 20
5.
a.
what is the location of the center?
(3)
b.
what is its radius?
(2)
For the circle given by:
x2 + 4x + y2 = 21
a.
what is the location of the center?
(5)
b.
what is its radius?
(5)
3-2
straight lines
6.
What are the slopes and the y-intercepts of the following lines?
a.
y = 3x + 4
(5)
b.
x
y =" +5
2
(5)
c.
4 x " 3y = 9
(5)
!
7.
! Write an equation for the line that
! a.
has a slope of −1 and a y-intercept of 2
(5)
b.
has a slope of ½ and a y-intercept of −1
(5)
2
c.
passes through the two points (−1,6) and (2,−2)
(5)
d.
passes through the two points (1,4) and (−6,−2)
(5)
e.
passes through the point (1,−3) and is parallel to the line:
y=
f.
x
"4
3
passes through the point (1,−3) and is perpendicular to the line:
!
y ="
3x
+4
2
3-3
functions
8.
Do the following define a function? Explain.
(5)
!
9.
a.
{(1,2),(1,3),(2,4),(3,6)}
(5)
b.
{(1,2),(2,3),(4,3),(6,4)}
(5)
c.
y = 3x2 + 2
(5)
d.
x = 3y2 + 2
(5)
e.
x = 3y + 5
(5)
f.
x2 + 3y2 = 5
(5)
What is the natural domain of the following?
!
a.
f (x) = 3x + 6
(5)
b.
f (x) = x " 2
(5)
c.
f (x) =
1
x +1
(5)
d.
f (x) =
3x + 2
x "2
(5)
e.
f (x) = x 2 + 1
!
!
10.
(5)
(5)
! What is the range of the following? (Assume a natural domain)
! a.
b.
!
!
f (x) = "2x + 3
(5)
f (x) = x + 3
(5)
3
c.
3-4
11.
f (x) = 2x 2 + 3
(5)
graphing functions
!
On what part of the domain is the following function increasing? decreasing?
(5)
g(x) = −2x2 + 4x + 5
12.
Is the function f(x)
= 3 a linear function? Why or why not?
13.
For the following quadratic functions, find the axis of symmetry, the vertex, the y-intercept,
and the x-intercepts, and sketch the graph of the function.
a.
f(x) = x2 + 6x − 5
(15)
b.
f(x) = −2x2 − 6x +3
(15)
3-5
combining functions
14.
For the functions:
f (x) = x 2 " 2,
g(x) = x + 1
what are the domains of f(x) and g(x)?
(6)
! b.
compute (f+g)(x) and its domain
(5)
c.
compute (f−g)(x) and its domain
(5)
d.
compute (fg)(x) and its domain
(5)
e.
compute (f/g)(x) and its domain
(5)
f.
compute (fog)(x) and its domain
(8)
g.
compute (gof)(x) and its domain
(8)
a.
15.
(5)
For the functions:
f (x) = 2x + 3,
g(x) = x "1
what are the domains of f(x) and g(x)?
(6)
! b.
compute (f+g)(x) and its domain
(5)
c.
compute (f−g)(x) and its domain
(5)
d.
compute (fg)(x) and its domain
(5)
a.
4
e.
compute (f/g)(x) and its domain
(5)
f.
compute (fog)(x) and its domain
(8)
g.
compute (gof)(x) and its domain
(8)
What is the effect on the graph of f(x) when you use
16.
a.
f(x+2)
(5)
b.
f(x) + 5
(5)
c.
−f(x)
(5)
d.
2•f(x)
(5)
3-6
inverse functions
17.
Are the following functions one-to-one? Explain
a.
f (x) = "2x " 6
(5)
b.
f (x) = 9 " x 2 , 0 # x # 3
(5)
c.
f (x) = x 4 + 2
(5)
! d.
f (x) = x 2 + 3, 0 " x
(5)
! e,
f (x) = 8
(5)
!
18.
! Find the inverse function for:
! a.
f (x) = 2x + 3
(10)
2x + 3
3x
(10)
b.
f (x) =
c.
f (x) = x + 1, "1 # x
!
(10)
19.
! What is the relationship between the graph of f(x) and f−1(x)?
(5)
20.
! What is f(x)of−1(x)?
(5)