A Graph y = A x 2 for A = 1 , 4 , and 1 4 simultaneously in the same coordinate system . B Graph y = A x 2 for A = − 1 , − 4 , and − 1 4 simultaneously in the same coordinate system. C Describe the relationship between the graph of h x = x 2 and the graph of G x = A x 2 for any real number A .
A Graph y = A x 2 for A = 1 , 4 , and 1 4 simultaneously in the same coordinate system . B Graph y = A x 2 for A = − 1 , − 4 , and − 1 4 simultaneously in the same coordinate system. C Describe the relationship between the graph of h x = x 2 and the graph of G x = A x 2 for any real number A .
Solution Summary: The author explains how to determine the maximum possible number of x-intercepts in a quadratic function.
A
Graph
y
=
A
x
2
for
A
=
1
,
4
,
and
1
4
simultaneously in the same coordinate system.
B
Graph
y
=
A
x
2
for
A
=
−
1
,
−
4
,
and
−
1
4
simultaneously in the same coordinate system.
C
Describe the relationship between the graph of
h
x
=
x
2
and the graph of
G
x
=
A
x
2
for any real number
A
.
Formula Formula A polynomial with degree 2 is called a quadratic polynomial. A quadratic equation can be simplified to the standard form: ax² + bx + c = 0 Where, a ≠ 0. A, b, c are coefficients. c is also called "constant". 'x' is the unknown quantity
7.
Define the sequence {b} by
bo = 0
Ել ։
= 2
8.
bn=4bn-1-4bn-2 for n ≥ 2
(a) Give the first five terms of this sequence.
(b) Prove: For all n = N, bn = 2nn.
Let a Rsuch that a 1, and let nЄ N. We're going to derive a formula for
Σoa without needing to prove it by induction. Tip: it can be helpful to use C1+C2+...+Cn
notation instead of summation notation when working this out on scratch paper.
(a) Take a a² and manipulate it until it is in the form Σ.a.
i=0
(b) Using this, calculate the difference between a Σ0 a² and Σ0 a², simplifying away the
summation notation.
i=0
(c) Now that you know what (a – 1) Σ0 a² equals, divide both sides by a − 1 to derive the
formula for
a².
(d) (Optional, just for induction practice) Prove this formula using induction.
3.
Let A, B, and C be sets and let f: A B and g BC be functions. For
each of the following, draw arrow diagrams that illustrate the situation, and then prove the
proposition.
(a) If ƒ and g are injective, then go f is injective.
(b) If ƒ and g are surjective, then go f is surjective.
(c) If gof is injective then f is injective. Make sure your arrow diagram shows that 9 does
not need to be injective!
(d) If gof is surjective then g is surjective. Make sure your arrow diagram shows that f
does not need to be surjective!
4.
5.
6.
Let X be a set and let f: XX be a function. We say that f is an involution if
fof idx and that f is idempotent if f f = f.
(a) If f is an involution, must it be invertible? Why or why not?2
(b) If f is idempotent, must it be invertible? Why or why not?
(c) If f is idempotent and x E range(f), prove that f(x) = x.
Prove that [log3 536] 5. You proof must be verifiable by someone who does not
have access to a scientific calculator or a logarithm table (you cannot use log3 536≈ 5.7).
Define the sequence {a} by a = 2-i for i≥ 1.
(a) Give the first five terms of the sequence.
(b) Prove that the sequence is increasing.
Chapter 2 Solutions
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
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