Glossary
Average Cost – The Average Cost (AC) for [latex]q[/latex] items is the total cost divided by [latex]q[/latex], or [latex]\frac{TC}{q}[/latex]. You can also talk about the average fixed cost, [latex]\frac{FC}{q}[/latex], or the average variable cost, [latex]\frac{TVC}{q}[/latex].
Average Revenue – The Average Revenue (AR) for [latex]q[/latex] items is the total revenue divided by [latex]q[/latex], or $$ \frac{TR}{q}. $$
axis of symmetry – a vertical line drawn through the vertex of a parabola, that opens up or down, around which the parabola is symmetric; it is defined by [latex]x=−\frac{b}{2a}.[/latex]
coefficients – The coefficients of a polynomial are the constant values that are multiplied by each variable term. For example, in the polynomial expression: [latex]3x^2 - 2x + 1[/latex] The coefficients are [latex]3, -2,[/latex] and [latex]1[/latex] . In this case, [latex]3[/latex] is the coefficient of the [latex]x^2[/latex] term, [latex]-2[/latex] is the coefficient of the [latex]x[/latex] term, and [latex]1[/latex] is the constant coefficient.
common logarithm – the exponent to which 10 must be raised to get [latex]x[/latex]; [latex]\log_{10}(x)[/latex] is written as [latex]\log (x)[/latex].
Composite Function – the new function formed by function composition, when the output of one function is used as the input of another
continuous growth – For all real numbers [latex]t[/latex], and all positive numbers [latex]a[/latex] and [latex]r[/latex], continuous growth or decay is represented by the formula
[latex]A(t) = a e^{rt}[/latex] where: [latex]a[/latex] is the initial value, [latex]r[/latex] is the continuous growth rate per unit time, and
[latex]t[/latex] is the elapsed time.
If [latex]r > 0[/latex], then the formula represents continuous growth. If [latex]r < 0[/latex], then the formula represents continuous decay. Cost – Your cost is the money you have to spend to produce your items.
degree – The degree of the polynomial is the highest power of the variable that occurs in the polynomial.
Demand – Demand is the functional relationship between the price [latex]p[/latex] and the quantity [latex]q[/latex] that can be sold (that is demanded). Depending on your situation, you might think of [latex]p[/latex] as a function of [latex]q[/latex] or of [latex]q[/latex] as a function of [latex]p[/latex].
Domain – The domain of a function is the set of all possible input values for which the function is defined.
exponential function – For any real number [latex]x[/latex], an exponential function is a function with the form [latex]f (x) = ab^x[/latex], where
[latex]a[/latex] is a non-zero real number called the initial value.
[latex]b[/latex] is any positive real number such that [latex]b \neq 1[/latex].
Fixed Cost – The Fixed Cost (FC) is the amount of money you have to spend regardless of how many items you produce. FC can include things like rent, purchase costs of machinery, and salaries for office staff. You have to pay the fixed costs even if you don’t produce anything.
Function – A function [latex]f[/latex] is a mathematical relation that maps elements from a set called the domain [latex]X[/latex] to a set called the codomain [latex]Y[/latex]. It associates each input element [latex]x[/latex] in the domain with a unique output element [latex]f(x)[/latex] in the codomain. The function can be defined using various notations, such as a formula, equation, or a set of rules. For example, we can define a function [latex]f[/latex] as follows:
[latex]f: X \rightarrow Y, \quad f(x) = \text{some expression involving } x[/latex]
where [latex]f(x)[/latex] represents the value obtained by applying the function [latex]f[/latex] to the input [latex]x[/latex]. The notation [latex]f: X \rightarrow Y[/latex] indicates that the function [latex]f[/latex] maps elements from the domain [latex]X[/latex] to the codomain [latex]Y[/latex]. The specific expression or rule defining the function may vary depending on the context and the nature of the function.
Function Notation – Consider a function [latex]f[/latex] defined by [latex]f(x) = 2x + 3[/latex], where [latex]x[/latex] is the input variable. It can also be written as [latex]y = 2x +3[/latex] and we say [latex]y[/latex] is a function of [latex]x[/latex]
horizontal asymptote – The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator.
Horizontal Compression – a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant [latex]b>1 [/latex]
horizontal intercept – Horizontal intercept (x-intercept) of a line: The point where the line intersects the horizontal axis (x-axis) on a graph.
Horizontal Line Test – The horizontal line test is a way to check if a function has a one-to-one relationship. Imagine drawing horizontal lines across the graph of a function. If any horizontal line intersects the graph at more than one point, then the function is not one-to-one because multiple inputs have the same output. However, if every horizontal line intersects the graph at most one point, then the function is one-to-one because each input has a unique output.
horizontal intercepts – Horizontal intercepts, also known as x-intercepts, refer to the points where a graph intersects the x-axis. These points have a y-coordinate of zero and can be found by setting y equal to zero and solving for x. In other words, they are the points on a graph where the function value is zero.
Horizontal line – a line defined by [latex]f(x)=b,[/latex]
where [latex]b[/latex]
is a real number. The slope of a horizontal line is [latex]0[/latex].
Horizontal Reflection – a transformation that reflects a function’s graph across the y-axis by multiplying the input by [latex]−1[/latex]
Horizontal Shift – a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input
Horizontal Stretch – a transformation that stretches a function’s graph horizontally by multiplying the input by a constant [latex]0
X-intercepts: roots, Zeros, Solutions
A quadratic equation may or may not touch x-axis. If it touches x-axis, it may touch x-axis at two points or only one point. These points are called x-intercepts. They can be found by solving quadratic equations.
leading coefficient - The leading coefficient is the coefficient of the leading term
leading term - The leading term is the term containing the highest power of the variable: the term with the highest degree.
linear function - a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line
logarithm - the exponent to which [latex]b[/latex] must be raised to get [latex]x[/latex] written by [latex]y = \log_b (x)[/latex]
Marginal Cost - The Marginal Cost (MC) at [latex]q[/latex] items is the cost of producing the next item.
Marginal Revenue - The Marginal Revenue (MR) at [latex]q[/latex] items is the revenue from producing the next item, $$ MR(q) = TR(q + 1) - TR(q). $$
natural logarithm - the exponent to which the number [latex]e[/latex] must be raised to get [latex]x[/latex];
[latex]\log_e(x)[/latex] is written as [latex]\ln (x)[/latex].
parallel lines - two or more lines with the same slope
perpendicular lines - two lines that intersect at right angles and have slopes that are negative reciprocals of each other
point-slope form - the equation for a line that represents a linear function of the form [latex]y−y_1=m(x−x_1)[/latex]
polynomial function - a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.
Profit - Your profit is what’s left over from total revenue after costs have been subtracted.
quadratic formula - [latex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/latex]
quadratic function - A polynomial of degree two which can be written in the form [latex]f(x) = ax^2 + bx + c[/latex]
Quadratics - A polynomial having degree 2 and it is a continuous function of the form [latex]f(x)=ax^2+bx+c[/latex] where [latex]a \neq 0[/latex]
Range - The range of a function is the set of all possible output values that the function can produce.
rational function - a function that can be written as the ratio of two polynomials
Revenue - Your revenue is the amount of money you actually take in from selling your products.
Slope - change in the vertical coordinates (rise) to the change in the horizontal coordinates (run) between two points on a line. [latex]m = \frac{y_2-y_1}{x_2-x_1}[/latex]
slope-intercept form - the equation for a line that represents a linear function in the form [latex]f(x)=mx+b[/latex]
standard form quadratic - Standard form of quadratic [latex]f(x)=ax^2+bx+c[/latex].
term - A term of the polynomial is any one piece of the sum, that is any [latex]a_i x^i[/latex]
Total Cost - The Total Cost (TC, or sometimes just C) for [latex]q[/latex] items is the total cost of producing them. It’s the sum of the fixed cost and the total variable cost for producing [latex]q[/latex] items
Total Variable Cost - The Total Variable Cost (TVC) for [latex]q[/latex] items is the amount of money you spend to actually produce them. TVC includes things like the materials you use, the electricity to run the machinery, gasoline for your delivery vans, maybe the wages of your production workers. These costs will vary according to how many items you produce.
vertex - the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function
vertex form of quadratic - vertex form of a quadratic [latex]f(x)=a(x-h)^2+k[/latex]
vertical - Vertical intercepts, also known as y-intercepts, refer to the points where a graph intersects the y-axis. These points have an x-coordinate of zero and can be found by setting x equal to zero and solving for y. In other words, they are the points on a graph where the input or independent variable has a value of zero, and the output or dependent variable takes a certain value.
vertical asymptote - The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero.
vertical Intercept - Vertical intercept (y-intercept) of a line: The point where a line intersects the vertical axis (y-axis) on a graph.
vertical line - a line defined by [latex]x=a[/latex], where [latex]a[/latex] is a real number. The slope of a vertical line is undefined.
Vertical Line Test - The vertical line test is a way to check if a graph represents a function. Imagine drawing a vertical line anywhere on the graph. If that line touches the graph at only one point, then the graph represents a function. However, if the line touches the graph at multiple points, then the graph does not represent a function
Vertical Reflection - a transformation that reflects a function’s graph across the x-axis by multiplying the output by −1
Vertical Shift - a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output
Vertical stretch - a transformation that stretches a function’s graph vertically by multiplying the output by a constant [latex]a>1[/latex]
