Math 1090
Chapter 3 Review
March 27th, 2015
The following is a list of questions covering chapter 3. The questions are
classified by sections, and each set is preceded by a brief summary of what
you should know from that section.
Section 3.1: Quadratic equations in one variable
A quadratic equation is an equations of the form ax2 + bx + c = 0, where a, b, c are real
numbers.
You should be able to:
• Solve quadratic equations as above by using the quadratic formula x =
√
−b± b2 −4ac
.
2a
• Solve quadratic equations by completing squares.
1. Solve the equation 15x2 + 24x − 12 = 0 using the quadratic formula.
2. Solve the equation −x2 − 8x + 7 = 0 using the quadratic formula.
3. Solve the equation x +
1
x
=
13
6
using the quadratic formula.
4. Solve the equation 5x2 − 3x − 1 = 0 by completing the square.
5. Solve the equation x2 +
3
2
= 1 by completing the square.
6. Solve the rational equation
1
x+4
+
1
x+3
7. Solve the rational equation
2x
x−6
=
x+1
x
=
1
3
Section 3.2: Parabolas: quadratic equations in 2 variables
A parabola is the set of points (x, y) that satisfy a quadratic equation in 2 variables
y = ax2 + bx + c
1. The sign of a determines whether the parabola is facing up (a > 0) or down (a < 0);
this also tells you whether the vertex is a maximum or a minimum.
2. The vertex is the highest/lowest point of a parabola and you can find its x-coordinate
b
applying the formula xv = − 2a
. Its y-coordinare is then given by yv = ax2v + bxv + c.
3. The axis of a parabola is a vertical line that passes through the vertex.
You should be able to:
• Given a quadratic equation in two variables, find the maximum/minimum, axis and
x-intercepts of the corresponding parabola.
• Use the previous information to sketch the graph of the parabola.
• Apply these notions to solve simple quadratic problems.
Find the vertex, the axis of symmetry, whether the parabola is concave up or down, the
x-intercepts and sketch the graph of the following parabolas
1. y = 3x2 − 2x + 1
2. y = 8(x − 1)2 + 2
3. y = −x2 − 2x − 2
ft
, its height above the ground
4. When an object is thrown upwards with a speed of 100 sec
2
is given by the function f (t) = −16t + 100t, where t is the time in seconds after it has
been thrown. At what time will the object reach its highest point? How hight will that
point be? How long does it take for the object to fall to the ground?
5. A rectangular playground is to be fenced in on all 4 sides. There is exactly 600 feet of
fencing for this playground. What dimensions give the largest area for the playground?
6. Given a parabola with vertex at (0, 2) that passes through the point (1, 8), find the
equations for the parabola.
Section 3.3: Quadratic business applications
We skipped this section and we will cover it in chapter 5
Section 3.4: Polynomial functions
A polynomial function is a function that can be written in the form
f (x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
, where an , . . . , a1 , a0 are real numbers. The degree is n and the leading coefficient is an . The
graph of a polynomial function of degree n will have n − 1 extreme points (e.g. a parabola
is given by a degree 2 polynomial and it has only one extreme point, which we called the
vertex). The sign of the leading coefficient an determines the shape of the graph (check your
notes).
You should be able to:
• Determine the degree, leading coefficient, x-intercepts, y-intercept and sketch the graph
of a polynomial function.
• Sketch the graph of a piecewise function.
1. Rewrite the following polynomial in standard form, write down its degree and its leading
coefficient and sketch its graph −3x2 + x4 + 1 + x.
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2. Rewrite the following polynomial in standard form, write down its degree and its leading
coefficient and sketch its graph −27y + y 2 − 31y 5 .
3. Rewrite the following polynomial in standard form, write down its degree and its leading
coefficient, find the x-intercepts and sketch its graph x(x + 3)(x − 2).
x + 5,
x ≤ −1
4. Consider the function f (x) =
What’s the value of the function
−2x + 8, x > −1
for x = −2, −1, 0, 1, 2.

 −2x, x ≥ 1
x2 ,
−1 ≤ x < 1 What’s the value of the function
5. Consider the function f (x) =

x + 3, x < −1
for x = −2, −1, 0, 1, 2.

x≥3
 3,
|x|,
−2 ≤ x < 3 What’s the value of the func6. Consider the function f (x) =

−(x + 1)2 x < −2
tion for x = −2, −1, 0, 1, 2.
Section 3.5: Rational functions
A rational function is a function that can be written as a quotient of polynomials, namely
f (x) =
p(x)
q(x)
where p(x), q(x) are polynomials. At the x-values where q(x) = 0, the rational function is
not defined: it blows up to ±∞. The vertical lines corresponding to those x-values are called
vertical asymptotes.
Sometimes, as x grows to ±∞, the function may collapse onto a horizontal line: those
lines are called horizontal asymptotes.
We said that the horizontal line y = a is a horizontal asymptote of a function f (x) if
lim f (x) = a
x→±∞
We saw that there are basically three possible situations, depending on the degree of the
polynomials p and q.
(a) If deg(p) > deg(q), then limx→±∞
p(x)
q(x)
(b) If deg(p) < deg(q), then limx→±∞
asymptote.
(c) If deg(p) = deg(q), then limx→±∞
is a horizontal asymptote.
= ±∞ and there is no horizontal asymptote.
p(x)
q(x)
p(x)
q(x)
= 0 and the line y = 0 (x-axis) is a horizontal
=
leading coefficient of p
leading coefficient of q
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and the line y =
leading coefficient of p
leading coefficient of q
For the each of the following rational functions:
1. Find the domain.
2. Find, if any, the vertical asymptote(s).
3. Find, if any, the horizontal asymptote(s).
4. Find the y-intercept.
5. Find the x-intercept.
6. Sketch the graph.
1. f (x) =
2
(x+2)2
2. f (x) =
3x+2
2x−1
3. f (x) =
10
x2 +2
4. f (x) =
2x−1
x2 +4x
5. f (x) =
x2 −4
x2 +6x+9
6. f (x) =
2x
x2 −4
Section 3.6: Transformations of graphs
In this section we saw how to obtain the graph of a function from the graph of another
one by performing simple transformations: shifting, reflecting with respect to the axes and
shrinking/stretching (see the tables on pages 193 and 194 of the textbook).
p
√
1. Graph the function f (x) = x and use it to graph the function g(x) = − 3(x − 2) + 4.
2. Graph the function f (x) = |x| and use it to sketch the graph of the function g(x) =
−2|x + 1| − 3.
3. Graph the function f (x) = x2 and use it to sketch the graph of g(x) = −(x + 1)2 + 3.
x
4. Graph the function f (x) = 2x and use it to sketch the graph of g(x) = − 12 + 1
Section 3.7: Combinations of functions
In this section we saw how to perform basic operations with functions. If f (x) and g(x)
are functions, we said that
• Addition: (f + g)(x) = f (x) + g(x)
• Substraction: (f − g)(x) = f (x) − g(x)
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• Multiplication: (f · g)(x) = f (x)g(x)
(x)
• Quotient: fg (x) = fg(x)
• Composition (f ◦ g)(x) = f (g(x)) (i.e. evaluate g at x, and then plug the resulting
expression into the expression for f .
For each question below, perform these operations:
(a) (f ◦ g)(x)
(b) g(h(x))
(c) (h ◦ g)
(d) f (g(h(x)))
(e) (f + g)(2)
(f) (hf )(3)
1. Perform all the operations above with the functions f (x) = 2x − 3, g(x) = x + 5 and
h(x) = x2
1
and
2. Perform all the operations above with the functions f (x) = 2x − 1, g(x) = x+5
h(x) = 3|x + 2|
√
3. Perform all the operations above with the functions f (x) = 2 x, g(x) = x3 and h(x) = x3
4. If f (x) = 2x2 − 1, find
f (x+h)−f (x)
.
h
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