College Algebra Summary

Recurring Themes

Algebra

The namesake of this course, “algebra”, is a rather overloaded word. In our context though it just boils down to symbolically manipulating equations. The ability to represent mathematical statements symbolically in terms of numbers and variables, as opposed to having to write it out in language, is a powerful tool if you can work with such representations. I.e. the question What number(s) when squared then tripled is equal to five more than half itself? is much easier to answer once you realize it’s the same as solving \(3x^2 = \frac{1}{2}x+5\) for \(x.\) And as a modern extension of this, it’s important to be able to accurately transcribe an algebraic expression for a number into a computer/calculator to approximate its decimal value.

• Skill

  Given an equation defined in terms of algebraic, exponential, and logarithmic operations, be able to solve the equation for any variable.

• Skill

  Be able effectively use technology to compute a decimal approximation of a number expressed in terms of algebraic, exponential, and logarithmic operations.

Analytic Geometry

This idea, a relatively modern extension to the study of algebra, is that every equation featuring variables has a geometric visualization. Specifically in this course, for a function \(f,\) the graph of \(f\) is the set of all points \((x,y)\) in two-dimensional space such that \({y = f(x).}\) The phrase analytic geometry refers to the study of functions via features of their graphs. This idea is helpful because it affords us two perspectives, an algebraic perspective and a geometric perspective, on any situation involving a function.

• Skill

  For any algebraic, exponential, or logarithmic function, given its formula be able to sketch the graph of the function, or given its graph be able to deduce a plausible formula for the function.

• Concept

  Understand how the features of the formula for an algebraic, exponential, or logarithmic function relate to the geometry of its graph, and vice-versa, and be able to view any facts about a function from the perspective of both its formula and its graph.

Modelling

One utility of mathematics accessible in this course is the use of functions to model phenomenon. That is if we’re measuring something over time \(t,\) it may be helpful to discover a function \(f\) such that \(f(t)\) approximates that measurement at time \(t.\) This function \(f\) affords us a better view into the phenomenon beyond what we can explicitly measure.

Given a collection of data \(\big\{(x_i, y_i)\big\},\) per type of function (linear, exponential, etc) we can use a computer/calculator to perform regression to compute the formula for that type of function that “best fits” the data. I.e. regression gives us a way to compute a function \(f\) such that \(f(x_i) \approx y_i\) for each data points, which then serves as a model for the phenomenon.

• Skill

  Be able to use technology to perform regression to commute the formula of a function that best fits data corresponding to a situation.

• Concept

  Understand how the algebraic or geometric properties of a function that models a situation should be interpreted in that context.

Mathematical Literacy

It’s also important to develop mathematical literacy in the more literal sense of literacy. Mathematical symbols are parts-of-speech and so have a proper place in the English language, and the exactness of mathematics demands mastery over vocabulary. Effectively communicating is an important skill in general, and so being able to incorporate mathematical elements into your writing and speech is a theme of this course.

• Skill

  Be able to communicate mathematical ideas effectively, using proper notation when necessary.

All of these themes listed more plainly as personal goals: be able to solve equations, know the geometry behind equations, know what the results of computations mean in context, and be able to effectively talk/write about all of it.

Mathematical Topics

Functions & Graphs

We start the course defining the main character of most all of modern mathematics: the function. A function, in general, is a correspondence between some set of inputs and some set of outputs, but in this class those sets will always be large subsets of the real numbers. Most quantities in our world are modelled by real numbers, and so phenomenon in our world can be usefully modelled by such functions. For a function \(f,\) given \(\square\) as input the output of \(f\) corresponding to that input is denoted \(f(\square).\) If \(f\) is defined by a formula, conventionally \(x\) is used in place of \(\square\) and \(f(x)\) is notation for the formula of \(f.\) The domain of a function \(f\) is the largest subset of the real numbers that may serve as inputs to \(f\), and the range of a function is the set of all its possible outputs.

• Concept

  Understand what a function is and the notation commonly used to denote information about functions.

And naturally here we introduce the graph of a function.

Linear Functions

After discussing functions in general, the first specific example that we analyze in depth are linear functions, named so for the shape of their graph. A linear function \(f\) has a formula of the template \(f(x) = mx+b.\) The two parameters \(m\) and \(b\) have clear interpretations in terms of a linear function’s graph: the number \(b\) is where the graph intersects the \(y\)-axis, and number \(m\) is the slope of the graph, the amount it changes in the \(y\) direction per change of \(1\) unit in the \(x\) direction. Beyond the understanding lines in view of the themes of the course, there is one skill worth mentioning specifically:

• Skill

  Given two points in space, be able to calculate an equation for the unique line that passes through those points.

Then lastly, before concluding our study of linear functions, we’ll practice analyzing multiple lines simultaneously in both the context of their algebra and geometry.

• Skill

  Given two lines in space, be able to calculate the point at which they intersects or determine that they are parallel.

Quadratic Functions

After lines we’ll study the next-simplest sort of algebraic function, quadratic polynomials. A quadratic polynomial function of \(x\) has a formula \(f(x)\)that may be written in any one of the following forms, \[ ax^2+bx+c \qquad a(x-h)^2+k \qquad a(x-r_1)(x-r_2) \] referred to as standard form, vertex form, and factored form respectively. The numbers \(r_1\) and \(r_2\) in the factored form, if they exist, are called the roots of \(f\) and satisfy \(f(r_1)=0\) and \(f(r_2)=0.\) Because \(f\) may not have real roots, it may not have a factored form.

• Skill

  Given the formula for a quadratic function, be able to re-write it any of these forms.

The graph of a quadratic function is called a parabola, and each of these forms encodes different information about the geometry to the corresponding parabola: the vertex of the parabola is located at \((h,k),\) the \(x\)-intercepts of the parabola (if they exist) are located at \((r_1,0)\) and \((r_2,0),\) the \(y\)-intercept of the parabola is located at \((0,c),\) and the values of \(a\) and \(b\) correspond to the rate at which the outputs are changing with respect to the inputs.

Miscellaneous Topics on Algebraic Functions

After studying linear and quadratic functions we introduce one more examples of function before discussing some miscellaneous topics about functions:

Power Functions. A power function of \(x\) has a formula of the template \(ax^b\) for any real number \(b.\) These functions are referred to as root functions in the case that \(0 \lt b \lt 1\) since, as a notational convention, \(x^{1/n} = \sqrt[n]{x}.\)

Piecewise-Defined Functions. All of the examples of the functions so far have been defined by a single formula, but this isn’t necessarily true of functions, A function defined in terms of cases, defined differently on different pieces of its domain, is called a piecewise-defined function.

Transformations of Functions. Focusing heavily for a moment on the general theme of analytic geometry, we study what basic algebraic augmentations to the formula of a function do to transform the shape of its graph.

• Skill

  Given the graph \(y = f(x),\) be able to sketch the graph \(y = af(cx+d)+b.\)

Algebraic Operations on Functions. We return briefly to the study of functions via their formulas, and define the operations \(+\) and \(-\) and \(\times\) and \(\div\) on functions based on those operations’ definitions on their outputs. We do define a single new operation on functions though, their composite, indicated with the symbol \(\circ,\) defined to be the function regarded as the two functions applied in serial, resulting from taking the outputs of the first function as the inputs of the second.

• Skill

  Given formulas for functions \(f\) and \(g\) be able to write down formulas for the functions \(f\pm g,\) \(f\times g,\) \(f \div g,\) and \(f \circ g.\)

Functions and their Inverses. We say a function is one-to-one if every output of the function corresponds to a unique input. We say a function \(f\) is invertible if there exists an function \(f^{-1},\) an inverse to \(f,\) such that \(f\big(f^{-1}(x)\big) = x\) and \(f^{-1}\big(f(x)\big) = x.\) It turns out a function is one-to-one if and only if it’s invertible.

• Skill

  Given a one-to-one function defined by a formula featuring a single instance of its independent variable, be able to write a formula for the inverse of that function.

Exponential & Logarithmic Functions

Next in the course we transcend algebra to talk about exponential functions of the form \(ab^x\) (notably the variable \(x\) is in the exponent) and their inverse logarithmic functions. We will spend a lot of time going over the mathematical mechanics of these functions before looking into their applications.

Exponential functions model measurements that are “growing at a rate proportional to their current size,” which makes them apt for studying population growth, disease spread, and finances. We look at this last example in depth, developing formulas for calculating compounding interest and for savings- and payout-annuities. Given a principle balance of \(P\) dollars invested at an annual interest rate of \(r\) compounded \(n\) times per year, the total balance after \(t\) years will be \[ P\bigg(1+\frac{r}{n}\bigg)^{nt}. \] If instead of being a one-time principle investment \(P\) was instead a regular deposit into an (initially empty) account made as often as the account earns interest, then the balance \(S\) after \(t\) years would be \[ S = P\left(\frac{\Big(1+\frac{r}{n}\Big)^{nt}-1}{\frac{r}{n}}\right) \,. \] If however the account has an initial balance of \(S\) dollars and each \(P\) is working towards paying off that balance, then the value of \(S\) after \(t\) years is \[ S = P\left(\frac{1-\Big(1+\frac{r}{n}\Big)^{-nt}}{\frac{r}{n}}\right) \,. \]

Briefly we introduce two other, more sophisticated functions in the “exponential” family, the logistic function and Gompertz function with these templates as function of \(x\) \[ \frac{C}{1 + a\mathrm{e}^{-bx}} \qquad \qquad C\alpha^{r^x} \] respectively. The motivation for their introduction is the desire to have bounded variations of the typical exponential functions; each of these functions ranges are bounded above by \(C.\)

Polynomial & Rational Functions

To conclude this course we address some of the patterns we noticed studying linear and quadratic functions earlier, and generalize them to polynomials of arbitrary degree. A polynomial, in full generality, is any function that can be defined in terms of applying only the operations \(\pm\) and \(\times\) to its input. Such a function’s formula has the template \[ a_nx^n + x_{n-1}x^{n-1} + \dotsb + a_2x^2 + a_1x + a_0\,. \] Much of the study of polynomials, like in the case of quadratics, comes down to finding their roots. We examine specific cases of degree-three (cubic) and degree-four (quartic) polynomials that we can factor, and we see how knowing a single root of a polynomial allows us to factor at least a linear term from it.

• Skill

  Given a polynomial \(f\) with root \(r\) be able to calculate the quotient \(q\) such that \[ f(x) = (x-r)\times q(x)\,. \]

This leads to a couple natural questions: Can all the roots be factored out from a polynomial one-at-a-time? How many roots does a given polynomial have? The answer to the first questions is yes, whereas the second is more complicated in general. After diving in the nuance of that second question for a bit, we introduce complex numbers, numbers of the form \(a+b\mathfrak{i}\) where \(a\) and \(b\) are real and \(\mathfrak{i}^2 = -1,\) and explain that their sole purpose is to provide a simple answer to that second question.

Any polynomial function with complex coefficients factors completely over the complex numbers.

And then we briefly introduce rational functions, which are any function that can be defined in terms of applying only the operations \(\pm,\) \(\times,\) and \(\div\) to its input. Any rational function can be written as the quotient of two polynomials. The roots of this “numerator” polynomial are referred to as the roots of the rational function, whereas the roots of the “denominator” are called poles of the rational function and correspond to vertical asymptotes in the function’s graph.