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ALEKS 1, 2 & 3 Algebra Answers Key
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Q. Rational Number
Ans: A number is rational if it can be written in the form a/b where a and b are integers
Q. Whole Number
Ans: They include 0 and natural numbers
Q. Natural Numbers
Ans: Counting numbers or positive integers
Q. Two inequalities joined by the word and or the word or form a __________ inequality. To graph a compound inequality,
Ans: Compound
Q. open circle on a number line
Ans: less than or greater than
Q. closed circle on a number line
Ans: less than or equal to, greater than or equal to
Q. Slope
Ans: It is a line in which a number measures how “steep” the line is
Q. if the slope is 0 then the line is:
Ans: Horizontal
Q. If the slope is a straight vertical line then it is:
Ans: Undefined
Q. Slope formula
Ans: (y₂- y₁) / (x₂- x₁) rise/run
Q. Relation
Ans: a set of ordered pairs
Q. Ordered Pairs
Ans:
The pair a,b is called ordered if it is known which element in the pair is the first and which is the second.
If a is the first element of the pair and b is the second, we use the notation (a,b)
Q. Set
Ans: A collection of objects. In mathematics, the objects of a set are called elements, members, or points
Q. Real Number
Ans: A number from one of the following sets: the natural numbers 1-etc or Integers ex -4 to etc or Rational numbers
Q. Rational Numbers
Ans: A number is rational if it can be written in the form a/b where a and b are integers
Q. Irrational Numbers
Ans:
A real number is irrational if it is not rational.
sqrt(2)
is irrational. In other words, there are no integers n and m such that n/m = sqrt 2 .
If a whole number is not a perfect square, then its square root is irrational.
Another irrational number is π
Q. Pi = 3.14159265
Ans:
Pi is a very important number that keeps appearing throughout mathematics. π(Pi) is an irrational number, so its decimal expansion never ends. That is why we name it with a symbol rather than just writing out its digits.
You May Be Interested: Algebra With Pizzazz Answer Key PDF
Q. Domain of a relation
Ans: the set of all first elements in the ordered pairs AKA x
Q. Range of a relation
Ans: the set of all second elements in the ordered pairs AKA y
Q. Function
Ans:
a rule that assigns to each element x in a set of A a unique element y in a set B
A function can be written as a set of ordered pairs (x, f(x)), where x varies over A. In a function, no two ordered pairs have the same first component
If A and B are sets of real numbers, then each pair (x, f(x)) gives the coordinates of a point. If we show all such points for a function we get the graph of the function
Q. Leading Coefficient of a polynomial
Ans: The leading term of a polynomial is the first term when the polynomial is in standard form.
Q. Degree of a polynomial
Ans: The degree of a polynomial is the highest degree of its terms.
Q. Binomial
Ans: A binomial is a type of polynomial that is a sum of only two monomials.
Q. Monomial
Ans: A monomial is a product of numbers and the powers of variables. EX: 2x^5
Q. Polynomial
Ans: A polynomial is a sum of monomials where each power is a positive whole number. Ex f(x) = -1 is a polynomial b/c this is a constant function (which is a polynomial with n = 0)
Q. The standard form of a polynomial
Ans: A polynomial in one variable is in standard form if the exponents on its variable decrease from left to right.
Q. Logorithm
Ans:
The base-a logarithm of x is the exponent to which a must be raised to get x.
The base-10 logarithm, which is log10, is written simply log. It is the common logarithm.
The base-e logarithm, which is loge, is denoted by ln. It is the natural logarithm
.The number e, sometimes called Euler’s number, is an irrational number with value 2.71828….
Q. Exponential Function
Ans: a function where x is in the exponent
Q. A relation cannot have the same _ component. A vertical line test proves whether or not its a function
Ans: first; x
Q. Average Rate of Change
Ans: same formula as slope
Q. Product
Ans: The result of multiplying two or more numbers or algebraic expressions
Q. Algebraic Expression
Ans: An expression or formula with numbers, variables, and operations
Q. For a product of expressions to equal 0, at least one of the expressions must equal __. For example A * B = 0 if and only if A = 0 or B = 0
Ans: 0
Q. interval notation
Ans: Including [ ] not including ( )
Q. The union of intervals uses the _ symbol. For example x < -1 OR x >=2
Ans: U
Q. Y intercept
Ans: The y coordinate of a point where the graph crosses the y axis or x = 0
Q. X intercept
Ans: The x coordinate of a point where the graph crosses the x axis or y = 0
Q. Slope intercept form
Ans: y = ax + b where y intercept is b
Q. Quadratic Function
Ans: if p and q are both 0’s of a quadratic function then we can write the equation as follows. Where a is a non-zero constant g(x) = a(x – p)(x – q)
Q. When a parabola is a function, it extends to the left and right. It also extends upward or downward _________.
Ans: forever
Q. Parabola
Ans: The graph of an equation y = ax^2 + bx + c, where a, b, and c are real numbers and a cannot be 0
Q. Axis of Symmetry formula for a parabola with equation y = ax^2 + bx + c. Plug this value x into the parabola equation to find y coordinate of the vertex
Ans: x = -b/2a
Q. Axis of Symmetry formula for a parabola with equation y = a(x-h)^2 + k. The axis of symmetry is the line x = h and the vertex is the point (h,k)
Ans: x = h and the vertex is the point (h,k)
Q. Zero of a Polynomial
Ans:
The number r is called a zero of the polynomial P(x) if p(r) = 0
such a number r can be a real number or a complex number that isn’t real.
R is a root or solution of the equation P(x) = 0 when r is a zero of the polynomial P(x)
Q. Factor Theorem
Ans:
If r is a zero of the polynomial P(x), then
x-r is a factor of
P(x)
Conversely, if
x-r is a factor of
P(x), then
r is a zero of the polynomial
P(x)
Q. Factor (verb)
Ans:
To factor, a given quantity means to express it as a product of factors ( with factors as a noun)
To factor, a whole number is to express it as a product of a whole number factors. For Ex. to factor 15 is to write it in form 5 * 3
To factor, a polynomial is to express it as a product of polynomial factors. For Ex. to factor x^2 + 3x + 2 is to write it in the form (x + 2)(x + 1)
Q. Multiplicity
Ans:
If a linear factor
x-r appears k
times in the product above, we say that
r is a zero of
P(x) of multiplicity
k.
A zero of multiplicity
2 is called a double zero.
We also get that, if
c is a zero of multiplicity
k, then
(x-c)^k is a factor.
Q. Rational Function
Ans: A function of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not the zero polynomial
Q. Domain of a rational function f(x) = p(x)/q(x) is x where q(x) cannot equal _
Ans: Zero
Q. Asymptote
Ans: a line is an asymptote of a curve if the curve approaches the line as we move along the line in one direction.
Q. Vertical Asymptote
Ans:
The line x = a
is a vertical asymptote of the graph of a function f if the values
f(x) either increase or decrease without bound as x
approaches
a from the right or from the left, that is, if
f(x) → ∞ or f(x) → −∞ as x → a+ or x → a− .
Q. Horizontal Asymptote
Ans:
The line
y = b is a horizontal asymptote of the graph of a function f if the values f(x) approach
b as the x values increase without bound or as the x values decrease without bound, that is,
f(x) → b as x → ∞ or x → −∞ .
Q. Local Minimum or Maximum
Ans:
The highest or lowest point within an area of the graph.
Cant be an endpoint, has to be defined on both left and right side
Somewhere in which the graph changes from increasing to decreasing or decreasing to increasing
Q. Absolute Minimum or Maximum
Ans: The highest or lowest point in the entire graph. It can be an endpoint
Q. Any zero with an even multiplicity is going to behave like a multiplicity of __
Ans: 2
Q. Any multiplicity that is odd and greater than one will behave like a multiplicity of _. Cubic
Ans: 3
Q. A horizontal asymptote is a line __ where the graph approaches the line as x approaches infinity and or as x approaches negative infinity
Ans: y = b. Tells us about end behavior. We can cross a horizontal asymptote
Q. A vertical asymptote is a line _ where the graph approaches negative infinity or positive infinity as the x gets close to a
Ans: x = a. Cannot be crossed. Happen where the function is not defined
Q. Parabola
Ans: The graph of a quadratic function of an equation y = ax^2 + bx + c, where a, b, and c are real numbers and a cannot equal 0.
Q. Axis of Symmetry
Ans: The line about which the parabola is symmetric for the equation y = ax^2 + bx + c it’s x = -b/2a
for the equation a(x – hh)^2 + k the axis of symmetry is x = h
Q. Vertex
The point where the parabola “turns around”
Q. Local Maxima
Ans:
A value at which f has a local maximum is the x-coordinate of a point on the graph where there is a local maximum.
A local maximum value of f is the y-coordinate of a point on the graph where there is a local maximum a function
f has a local maximum at
a if the value of
f at a is greater than or equal to any value of f n
Q. Minima
Ans:
a function could have no local maxima, one local maximum, or more than one local maximum. Same is true for local minima
a function f has a local minimum at b if the value of f at b is less than or equal to any value of f near b.
A value f(b) is a local minimum of a function f if there is an open interval containing b such that f(b) <= f(x) for all x in the open interval
Q. Absolute Maximum of f
Ans:
The greatest y coordinate of any point on the graph of f
if a “hole” in the graph of f has a greater y coordinate than any point on the graph of f then the function does not have an absolute maximum
if the greatest point is going towards infinity its none
Q. Absolute Minimum of f
Ans:
The least y coordinate of any point on the graph of f
if a “hole” in the graph of f has a lesser y coordinate than any point on the graph of f then the function does not have an absolute minimum
if the greatest point is going towards infinity/-infinity its none
Q. “Hole”
Ans: shown as a hollow dot, a point that is not on the graph of f
Q. Three transformations of the graph of a function:
Ans: translations, reflections & expansions/contractions
Q. Translations
Ans: if each of the points of a graph is moved the same distance in the same direction, we say that the resulting graph is a translation of the original.
Q. Reflection about the y axis
Ans:
if f is a function whose domain and range are subsets of the set of real numbers, then the reflection of the graph of f about the y-axis is the reflection of every point on the graph of f about the y axis.
The reflection of the graph of f about the y axis is the graph of the function g(x) = f(-x)
Q. reflection about the x axis
Ans:
If f is a function whose domain and range are subsets of the set of real numbers, then the reflection of the graph of f about the
x-axis is the reflection of every point on the graph of
f about the x-axis.
The reflection of the graph of f about the
x-axis is the graph of the function g(x)=f(-x)
Q. Expansion/Contractions: Vertical Expansion and Contractions
Ans:
Suppose that f is a function whose domain and range are subsets of the set of real numbers:
– if c is a number greater than 1, then the graph of g(x) = cf(x) Is a vertical expansion, also called a vertical stretching, of the graph f
– if c is a number between 0 and 1, then the graph of g(x) = cf(x) Is a vertical contraction, also called a vertical shrinking, of the graph of f
Q. Horizontal Expansions of Contractions
Ans:
Suppose that f is a function whose domain and range are subsets of the set of real numbers:
-if c is a number greater than 1, then the graph of g(x) = f(cx) Is a horizontal contraction, also called a horizontal shrinking of the graph of f
-if c is a number between 0 and 1 then the graph of g(x) = f(cx) is a horizontal expansion, also called horizontal stretching of the graph of f
Q. Horizontal Translation: To graph y = |x – c|, we shift the graph of y = |x| to the _______ c units
Ans: right
Q. Horizontal Translation: To graph y = |x + c|, we shift the graph of y = |x| to the _______ c units
Ans: left
Q. Vertical translations: To graph y = |x| + c, we shift the graph of y = |x| ______ c units
Ans: upward
Q. Vertical translations: To graph |x| -c, we shift the graph of y =|x| _______ c units
Ans: downward
Q. A graph is symmetric with respect to the x-axis if ________.
Ans: for every point (x, y) on the graph, the point (x, -y) is also on the graph
Q. A graph is symmetric with respect to the y-axis if __.
Ans: for every point (x, y) on the graph, the point (-x, y) is also on the graph
Q. A graph is symmetric with respect to the origin if _.
Ans: or every point (x, y) on the graph, the point (-x, -y) is also on the graph
Q. A leading coefficient (a > 0) gives a parabola that opens ________
Ans: positive, upward
Q. A _______ leading coefficient (a > 0) gives a parabola that opens _______
Ans: positive, upward
Q. A ______ leading coefficient (a < 0) gives a parabola that opens ________
Ans: negative, downward
Q. A _______ parabola has a leading coefficient a, closer to 0
Ans: wider
Q. A _______ parabola has a leading coefficient a, farther from 0
Ans: narrower
Q. Vertical Expansions and Contractions
Ans:
If c is a number greater than 1, then the graph of g(x) = cf(x) a vertical expansion, also called a vertical stretching, of the graph of f.
If c is a number between 0 and 1, then the graph of g(x) = cf(x) a vertical contraction, also called a vertical shrinking, of the graph of f.
In both cases, for a given x coordinate, the
y coordinate of the point on the graph of
g is c times the y coordinate of the point on the graph of f.
Q. Horizontal Expansion and Contractions
Ans:
If c is a number greater than 1, then the graph of g(x) = f(cx) a horizontal contraction, also called a horizontal shrinking, of the graph of f.
If c is a number between 0 and 1, then the graph of g(x) = f(cx) a horizontal expansion, also called a horizontal stretching, of the graph of f.
In both cases, for a given y coordinate, the
x coordinate of the point on the graph of
g is 1/c times the x
coordinate of the point on the graph of f
Q. Reflection across x-axis
Ans: -f(x) the negative is on the outside
Q. Reflection across the y-axis
Ans: f(-x) the negative is on the inside
Note: Algebra questions in ALEKS are prepared by professional mathematicians whose areas of expertise give the program the ability to ask questions that are both challenging and beneficial to the learning process.
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