MAT 120 Exam 1 Concepts

All graphs of exponential functions will pass through the point ( , ).

(0,1)

If ( a , b ) is a point on the graph of a one-to-one function f, then ( , ) is a point on the graph of the inverse of f.

(b,a)

Common logarithms are logarithms with base ________. Natural logarithms are logarithm with base_______

10 e

Shapes of Odd Degree Polynomial Graphs All odd degree polynomials cross the x axis _________ . If the polynomial starts negative it ends positive whenever the leading coefficient is _______. If the polynomial starts positive it ends negative whenever the leading coefficient is ________ .

at least one time positive negative

The ________or rectangular coordinate system consists of two real number lines, one horizontal and one vertical, that intersect at their origins. The two number lines are called the _________axis and the ___________axis. Together, they are referred to as the coordinate axes. The horizontal axis is usually referred to as the _________ axis and the vertical axis as the _________axis. The coordinate axes divide the plane into four parts called _________, which are numbered counterclockwise using Roman numerals from I to IV, starting with the upper right quadrant.

cartesian coordinate system horizontal vertical x y quadrants

The formula for _______is A = P ( 1 + r /m ) m ⋅ t. The formula for is________ A = P e r ⋅ t

compound interest continuous compound interest

Functions in the form of y = f ( x ) = c ⋅ e k t, are commonly used to model popular growth and radioactive decay. The constant c, represents the ______amount. The constant k, represents the ________(percentage rate per unit of time in decimal form).

constant relative growth rate

The ________ for a function f ( x ) for x and x + h in the domain of f, with h ≠ 0 is f ( x + h ) − f ( x )/ h. The difference quotient finds the _________of the line between two points on the graph of a function.

difference quotient slope

Independent and Dependent Variables Input values for functions are ________values and output values for functions are _______ values. Function equations often use the variable, x as the [domain] variable and the variable, y as the [range] variable. The input variable is called an ________variable. The output variable is called a _________variable. If a function is specified by an equation and the domain is not indicated, we assume that the domain is the set of all real-number replacements of the independent variable that produce real values for the dependent variable. The independent variable represents [domain] values. The dependent variable represents [range] values. The [range] of a function is the set of all outputs corresponding to input values.

domain range independent dependent

The _______of a logarithmic function is the set of all positive real numbers. The ________of a logarithmic function is the set of all real numbers.

domain range

A special exponential function is one that has base ______and base 1/ ______.

e e

Functions Specified by Equations If in an equation in two variables, we get __________(value for the dependent variable) for each ________(value for the independent variable), then the equation specifies a function. The _______of such a function is just the graph of the specifying equation. If we get more than one out put for a given input, the equation _______ specify a function.

exactly one out input graph does not

Let b > 0, b ≠ 1 be a real number. Then f ( x ) = b x defines an _______function. The constant b is called the_________ .

exponential base

Profit-loss Analysis Companies use profit-loss analysis to support decisions regarding the pricing of products and appropriate levels of production to maximize company profit. A manufacturing company has costs, C, which include _______costs (plant overhead, product design, setup, and promotion) and _________costs (costs that depend on the number of items produced.) The ________(income) for a company, R, is the amount of money the company receives from selling its product. _________is equal to revenue minus cost.

fixed variable revenue profit

A mathematical _________is a correspondence between two sets of elements such that to each element in the first set, there corresponds one and only one element in the second set. The first set in such a correspondence is called the _________of the function. The second set in such a correspondence is called the __________of the function. A key concept for functions is the requirement that each domain element corresponds with one and only one range element.

function domain range

A ________of a rational function is a line of the form y = k which the graph of the function approaches but does not cross as both x increases and decreases without bound.

horizontal asymptote

If b > 1, then b x _______as x increases. If 0 < b < 1, then b x _______as x increases

increases decreases

If f is a one-to-one function, then the _______of f is the function formed by interchanging the independent and dependent variables for

inverse

When a new function is formed by performing an operation on a given function, the graph of the new function is called a ________of the graph of the original function

inverse

If m and b are real numbers with m ≠ 0, then the function f ( x ) = m x + b is a _______function.

linear

A polynomial of degree 0 is a constant function. A polynomial of degree 1 is a ______function. A polynomial of degree 2 is a ________ function.

linear quadratic

The inverse of an exponential function is called a _______function.

logarithmic

R<C then R>C then R=C then

loses money makes profit breaks even

A function is said to be ______if each range value corresponds to exactly one domain value.

one-to-one

The Fundamental Theorem of Analytic Geometry There is a __________between the points in a plane and the elements in the set of all ordered pairs of real numbers. Each point in the Cartesian coordinate system corresponds to exactly one ordered pair of real numbers. Each ordered pair of real numbers corresponds to exactly one point in the Cartesian coordinate system.

one-to-one correspondence

The graph of a quadratic function f ( x ) = a x 2 + b x + c is a ________ opening upward for a > 0 and ________for a < 0.

parabola downward

_________ is a process where ordered pairs of points that solve an equation are found. The points are plotted on a grid and then connected with a smooth curve.

point by point graphing

A ________function is a function that can be written in the form a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 for a nonnegative integer n, called _______the of the polynomial. The coefficients a 0 , a 1 , ... , a n are real numbers with a n ≠ 0. The coefficient of the term of highest degree is called the ________coefficient.

polynomial degree leading

Vertical Shifts The graph of y = f ( x ) + k can be obtained from the graph of y = f ( x ) by vertically translating (shifting) the graph of the latter upward k units if k is ______ and downward | k | units if k is ________ .

postive negative

If a, b, and c are real numbers with a ≠ 0, then the function f ( x ) = a x 2 + b x + c is a __________function.

quadratic

Let b > 0, b ≠ 1 be a real number, and let f ( x ) = b x. The _______of f is the set of all real numbers. The ________of f is the set of all positive real numbers.

range domain

A _______function is any function that can be written in the form f ( x ) = n ( x ) d ( x ), where n ( x )and d ( x )are polynomials and d ( x ) ≠ 0.

rational

The ________is equal to the number of items sold multiplied by the price per item.

revenue

If ( x 1 , y 1 ) and ( x 2 , y 2 ) are two points on a line with x 1 ≠ x 2, then the _______of the line is m = y 2 − y 1 x 2 − x 1.

slope

A _________ to an equation in two variables is an ordered pair of numbers that when substituted into the equation result in a true statement.

solution

Stretches and Shrinks The graph of y = A f ( x ) can be obtained from the graph of y = f ( x ) by multiplying each ordinate value of the latter by A and graphing the points with these new ordinate values. Values where A > 1 result in a vertical _________ of the graph of y = f ( x ). Values where 0 < A < 1 result in a vertical ________of the graph of y = f ( x ). If A = − 1, the result is a _________ of the graph in the x -axis with no vertical stretching or compression

stretch compression reflection

Horizontal Shift The graph of y = f ( x + h ) can be obtained from the graph of y = f ( x ) by horizontally translating (shifting) the graph of the latter h units _______ if h is positive and | h | units _________ if h is negative.

to the left to the right

Given a quadratic function f ( x ) = a x 2 + b x + c with a ≠ 0, completing the square can transform the function to the _______form f ( x ) = a ( x − h ) 2 + k. The coordinates for the ________are ( h , k ). f ( h ) = k is the _______if a > 0 and the ________if a < 0.

vertex vertex minimum maximum

The graph of y = f ( x ) + k gives a _____transformation of the graph of y = f ( x ).

vertical

The x-axis is a ______-asymptote for every exponential function.

vertical

A ________of a rational function f ( x ) is a line of the form x = h which the graph of the function approaches but does not cross.

vertical asymptote

Suppose f ( x ) = n ( x ) /d ( x )is a rational function, where n ( x ) and d ( x ) are polynomials. If n ( x ) and d ( x ) have no real zeros in common and c is a real number such that d ( c ) = 0, then the line ______ is a ________of the graph of f. If n ( x ) and d ( x ) have one or more real zeros in common, ________ . If the degree of n ( x ) is less than the degree of d ( x ), then _______ is the horizontal asymptote . If the degree of n ( x ) is equal to the degree of d ( x ), then _______ is the ________, where ________ is the leading coefficient of n ( x ) and _______ is the leading coefficient of d ( x ). If the degree of n ( x ) is greater than the degree of d ( x ), then there is no _________

x=c vertical asymptote cancel the common factors y=0 horizontal asymptote y=a/b horizontal asymptote a b horizontal asymptote

Given a quadratic function in vertex form f ( x ) = a ( x − h ) 2 + k, the axis of symmetry is ________ The domain is the set of all real numbers . The range is ( − ∞ , k ]if _________and [ k , ∞ ) if a > 0 .

x=h a<0