Unit 1 Lesson 2 - In this recording I review some of the important concepts from this lesson including simplifying expressions, using rules of exponents, and solving quadratic equations by factoring, completing the square and the quadratic formula.
Unit 1 Lesson 3 - In this recording we explain relations and functions, explore the vertical line test and distinguish between continuous and discrete functions.
Unit 1 Lesson 4 - In this recording we introduce the idea of complex numbers. We then demonstrate how to find their magnitude, and multiply subtract them. Finally, we explore the Mandelbrot set as an enrichment activity.
Unit 2 Lesson 1 - In this video I introduce set builder and interval notation. We then use that to express the domain and range of functions given their graphs and using the two "don'ts." Don't divide by 0 and don't take the square root of a negative.
Unit 2 Lesson 2 - Here I demonstrate adding, subtracting, multiplying and dividing functions, as well as determining domain restrictions.
Unit 2 Lesson 3 - Composition of functions is explained in this introduction video. I use composition to create a function that turns a given Fahrenheit temperature into Kelvin.
Unit 2 Lesson 4 - Here we introduce the idea of an inverse relation or function as switching input and output, or x and y. We then expand it to y = x symmetry on a graph and find the inverse of several functions. Mention is made of one-to-one functions and domain restrictions.
Unit 2 Lesson 4 Domain Restrictions - For a function to have an inverse, it must be one-to-one. It is possible to make a function that is not one-to-one such by restricting its domain. In this video I take a quadratic function and restrict its domain so that we can find its inverse function.
Unit 2 Lesson 5 - In this LiveLesson we distinguish between finding an inverse and verifying or proving two functions are inverses. The distinction is stark, and many students lose points on assessments over it. Take note of the contents of this video.
Unit 2 Lesson 6 - Here we review what an inverse function is. We then graph several relations and functions in Desmos, noting the reflection across the line y = x. Finally, we sketch the graph of a given function from its graph alone.
Unit 3 Lesson 1 - In this recording I use Desmos to graph a piecewise function and then analyze where the function is increasing, decreasing, or constant. I then find the x and y intercepts. Finally, I formally define what it means for a function to be increasing, decreasing, or constant.
Unit 3 Lesson 2 - Here I define what is meant by even function and odd function. I then demonstrate what that means graphically.
Unit 3 Lesson 3 - In this video I demonstrate finding vertical and horizontal asymptotes for a rational function. We also explain the implications of a removable (cancelling) discontinuity to the asymptotes. Then I explain the thinking behind end behavior for polynomial functions.
Unit 3 Lesson 4 - In this video I explain the various types of discontinuities: hole (removable), jump, asymptotic (infinite). I then give specific examples of when you might run into each.
Unit 3 Lesson 5 - This recording presents the concept of a parent function and transformations on that function. Then it introduces the linear parent function, y = x, the absolute value parent function y = |x|, and the reciprocal parent function y = 1/x.
Unit 3 Lesson 6 - This recording introduces the functions families: exponential function, logarithmic function, power function and root function. Note these are in inverse pairs.
Unit 3 Lesson 7 - In this video I demonstrate transformations on functions by applying them to the sine function. These transformations may be applied to any of the function families, starting with a given parent function.
Unit 4 Lesson 1 - In this video I define polynomial function and the terms associated with it. Then I use Desmos to explore their graphs, including end behavior, maximum curves and local and global minima and maxima, also called extrema.
Unit 4 Lesson 2 - In this video I use Desmos to demonstrate the behavior of odd and even roots or zeros of a function. I then explain the rational roots theorem for polynomial functions and how to use the Intermediate Value Theorem (IVT) to determine where roots or zeros must exist. Finally, I explain Descarte's rule for positive and negative zeros of a polynomial function.
Unit 4 Lesson 3 - In this video I use the rational roots theorem, long division and synthetic division to find the factors and zeros of a polynomial function.
Unit 4 Lesson 4 - In this video I use the rational roots theorem and synthetic division to factor a polynomial function as far as it will go. I then use the quadratic formula to find the pair or complex roots for the function and demonstrate that complex roots always come in conjugate pairs.
Unit 4 Lesson 5 - In this video I demonstrate how to find vertical asymptotes, hole discontinuities and end behavior of rational functions using polynomial division or factoring and long division.
Unit 4 Lesson 6 - In this video I review how to add, subtract, multiply and divide fractions. I then apply those rules to operations on rational functions. I briefly discuss what it means for operations on rational functions to be closed.
Unit 5 Lesson 1 - In this video I explain what an exponential function is and compare it to a linear function. I then demonstrate the shape of an exponential functions graph for different parameter values. I demonstrate how to use exponent rules to write a function in exponential form and finally introduce the natural base, e = 2.718282....
Unit 5 Lesson 2 - In this video I explain what a logarithmic function is. I then use the definition to convert between logarithmic and exponential forms of the same function. Finally, I demonstrate how to calculate logarithms without a calculator.
Unit 5 Lesson 3 - In this video I graph y = 2^x and y = log_2 (x) on the same axis to demonstrate their inverse properties. I identify the factors that help determine the difference between exponential and logarithmic functions. I then use Desmos to demonstrate what happens to both functions as we change the base to 3. Finally, I demonstrate how to translate, reflect and dilate the logarithmic function.
Unit 5 Lesson 4 - In this video I present three rules of logarithms. I then use those rules to expand and contract expressions. Then I use three different techniques to solve logarithmic equations.
Unit 5 Lesson 5 - In this video I solve three exponential equations. I solve one by rewriting it as a logarithmic equation. I solve another by taking the log of both sides. I solve the last by rewriting it in quadratic form and using y-substitution.
Unit 5 Lesson 6 - In this video I explain what a piecewise function is. I demonstrate how to enter a piecewise function in Desmos. I then demonstrate how to find coefficient values that make a function continuous.
Unit 6 Lesson 1 - In this video I explain the origins of the distance formula and use it to find the distance between two points. I then explain the origins of the midpoint formula and use it to calculate the midpoint between two points.
Unit 6 Lesson 2 - In this video I use pictures to demonstrate what conic sections are. I then present the general form of a conic section and use the discriminant to determine what kind of conic section it is.
Unit 6 Lesson 3 - In this video I demonstrate how to determine what conic section you have by taking the discriminant of the general form. I then convert the general form to the standard form for a circle, identify the center and radius of the circle and graph it.
Unit 6 Lesson 4 - In this video I demonstrate how to determine an ellipse from the general form. I then demonstrate how to convert general form to standard form. Finally, I demonstrate how to graph the equation from the standard form and find the coordinates of the foci.
Unit 6 Lesson 5 - In this video I demonstrate how to determine a parabola from the general form. I then demonstrate how to convert general form to standard form. Finally, I demonstrate how to graph the parabola from the general form, including the focus and directrix.
Unit 6 Lesson 6 - In this video I demonstrate how to determine if a polynomial in general form is a hyperbola. I then use completing the square to convert it to the standard form for a hyperbola. Finally, I demonstrate how to graph it and find its foci.
Unit 7 Lesson 1 - In this video I introduce the concept of a sequence with its notation. I then explain the difference between explicit and recursive sequence definitions, working out the first 5 terms of each.
Unit 7 Lesson 2 - In this video I demonstrate how to write the explicit formula and the recursive formulas for any given arithmetic sequence.
Unit 7 Lesson 3 - In this video I review what an arithmetic sequence is, and define a sequence both recursively and explicitly. I then demonstrate how to find the first term given the common difference and a specified term, how to find any other term asked for, and how to find missing terms between two given terms in an arithmetic sequence.
Unit 7 Lesson 4 - In this video I explain the difference between a sequence and a series. I then use a specific example to derive the formula for the partial sum of an arithmetic series.
Unit 7 Lesson 5 - In this video I introduce sigma notation for the partial sum of series in general. I then demonstrate how to write an arithmetic series in sigma notation. I present the various known summation formulas for k, i, i^2, and i^3 and use the properties of summation to evaluate a given finite series.
Unit 7 Lesson 6 - In this video I review the recursive and explicit definitions of an arithmetic sequence. I then present an example of a geometric sequence and define it recursively and explicitly. Finally, I demonstrate how to find a previous term given the nth term and the common ratio, and how to find missing terms between two given terms.
Unit 7 Lesson 7 - In this video I demonstrate how to find the first term of a geometric sequence given two terms, as well as finding any subsequent terms. I then demonstrate how to find missing terms between any two given terms.
Unit 7 Lesson 8 - In this video I explain how it is that adding an infinite number of terms can result in a fine sum or total. I then give the harmonic series as an example of a divergent series and that a geometric series in which |r| < 1 is always convergent, having a finite sum.
Unit 7 Lesson 9 - In this video I review what a geometric series is. I then present the idea of a finite sum of a geometric series. I then derive the formula for the sum of a finite geometric series and use it to find the sum of the first 100 terms of the series 5 + 10 + 20 + 40 + 80 + ... .
Unit 7 Lesson 10 - In this video I derive the formula for an infinite geometric series from that for a finite series. I then use the formula to find the sum of an infinite series.
Unit 1 Lesson 3 - In this recording we explain relations and functions, explore the vertical line test and distinguish between continuous and discrete functions.
Unit 1 Lesson 4 - In this recording we introduce the idea of complex numbers. We then demonstrate how to find their magnitude, and multiply subtract them. Finally, we explore the Mandelbrot set as an enrichment activity.
Unit 2 Lesson 1 - In this video I introduce set builder and interval notation. We then use that to express the domain and range of functions given their graphs and using the two "don'ts." Don't divide by 0 and don't take the square root of a negative.
Unit 2 Lesson 2 - Here I demonstrate adding, subtracting, multiplying and dividing functions, as well as determining domain restrictions.
Unit 2 Lesson 3 - Composition of functions is explained in this introduction video. I use composition to create a function that turns a given Fahrenheit temperature into Kelvin.
Unit 2 Lesson 4 - Here we introduce the idea of an inverse relation or function as switching input and output, or x and y. We then expand it to y = x symmetry on a graph and find the inverse of several functions. Mention is made of one-to-one functions and domain restrictions.
Unit 2 Lesson 4 Domain Restrictions - For a function to have an inverse, it must be one-to-one. It is possible to make a function that is not one-to-one such by restricting its domain. In this video I take a quadratic function and restrict its domain so that we can find its inverse function.
Unit 2 Lesson 5 - In this LiveLesson we distinguish between finding an inverse and verifying or proving two functions are inverses. The distinction is stark, and many students lose points on assessments over it. Take note of the contents of this video.
Unit 2 Lesson 6 - Here we review what an inverse function is. We then graph several relations and functions in Desmos, noting the reflection across the line y = x. Finally, we sketch the graph of a given function from its graph alone.
Unit 3 Lesson 1 - In this recording I use Desmos to graph a piecewise function and then analyze where the function is increasing, decreasing, or constant. I then find the x and y intercepts. Finally, I formally define what it means for a function to be increasing, decreasing, or constant.
Unit 3 Lesson 2 - Here I define what is meant by even function and odd function. I then demonstrate what that means graphically.
Unit 3 Lesson 3 - In this video I demonstrate finding vertical and horizontal asymptotes for a rational function. We also explain the implications of a removable (cancelling) discontinuity to the asymptotes. Then I explain the thinking behind end behavior for polynomial functions.
Unit 3 Lesson 4 - In this video I explain the various types of discontinuities: hole (removable), jump, asymptotic (infinite). I then give specific examples of when you might run into each.
Unit 3 Lesson 5 - This recording presents the concept of a parent function and transformations on that function. Then it introduces the linear parent function, y = x, the absolute value parent function y = |x|, and the reciprocal parent function y = 1/x.
Unit 3 Lesson 6 - This recording introduces the functions families: exponential function, logarithmic function, power function and root function. Note these are in inverse pairs.
Unit 3 Lesson 7 - In this video I demonstrate transformations on functions by applying them to the sine function. These transformations may be applied to any of the function families, starting with a given parent function.
Unit 4 Lesson 1 - In this video I define polynomial function and the terms associated with it. Then I use Desmos to explore their graphs, including end behavior, maximum curves and local and global minima and maxima, also called extrema.
Unit 4 Lesson 2 - In this video I use Desmos to demonstrate the behavior of odd and even roots or zeros of a function. I then explain the rational roots theorem for polynomial functions and how to use the Intermediate Value Theorem (IVT) to determine where roots or zeros must exist. Finally, I explain Descarte's rule for positive and negative zeros of a polynomial function.
Unit 4 Lesson 3 - In this video I use the rational roots theorem, long division and synthetic division to find the factors and zeros of a polynomial function.
Unit 4 Lesson 4 - In this video I use the rational roots theorem and synthetic division to factor a polynomial function as far as it will go. I then use the quadratic formula to find the pair or complex roots for the function and demonstrate that complex roots always come in conjugate pairs.
Unit 4 Lesson 5 - In this video I demonstrate how to find vertical asymptotes, hole discontinuities and end behavior of rational functions using polynomial division or factoring and long division.
Unit 4 Lesson 6 - In this video I review how to add, subtract, multiply and divide fractions. I then apply those rules to operations on rational functions. I briefly discuss what it means for operations on rational functions to be closed.
Unit 5 Lesson 1 - In this video I explain what an exponential function is and compare it to a linear function. I then demonstrate the shape of an exponential functions graph for different parameter values. I demonstrate how to use exponent rules to write a function in exponential form and finally introduce the natural base, e = 2.718282....
Unit 5 Lesson 2 - In this video I explain what a logarithmic function is. I then use the definition to convert between logarithmic and exponential forms of the same function. Finally, I demonstrate how to calculate logarithms without a calculator.
Unit 5 Lesson 3 - In this video I graph y = 2^x and y = log_2 (x) on the same axis to demonstrate their inverse properties. I identify the factors that help determine the difference between exponential and logarithmic functions. I then use Desmos to demonstrate what happens to both functions as we change the base to 3. Finally, I demonstrate how to translate, reflect and dilate the logarithmic function.
Unit 5 Lesson 4 - In this video I present three rules of logarithms. I then use those rules to expand and contract expressions. Then I use three different techniques to solve logarithmic equations.
Unit 5 Lesson 5 - In this video I solve three exponential equations. I solve one by rewriting it as a logarithmic equation. I solve another by taking the log of both sides. I solve the last by rewriting it in quadratic form and using y-substitution.
Unit 5 Lesson 6 - In this video I explain what a piecewise function is. I demonstrate how to enter a piecewise function in Desmos. I then demonstrate how to find coefficient values that make a function continuous.
Unit 6 Lesson 1 - In this video I explain the origins of the distance formula and use it to find the distance between two points. I then explain the origins of the midpoint formula and use it to calculate the midpoint between two points.
Unit 6 Lesson 2 - In this video I use pictures to demonstrate what conic sections are. I then present the general form of a conic section and use the discriminant to determine what kind of conic section it is.
Unit 6 Lesson 3 - In this video I demonstrate how to determine what conic section you have by taking the discriminant of the general form. I then convert the general form to the standard form for a circle, identify the center and radius of the circle and graph it.
Unit 6 Lesson 4 - In this video I demonstrate how to determine an ellipse from the general form. I then demonstrate how to convert general form to standard form. Finally, I demonstrate how to graph the equation from the standard form and find the coordinates of the foci.
Unit 6 Lesson 5 - In this video I demonstrate how to determine a parabola from the general form. I then demonstrate how to convert general form to standard form. Finally, I demonstrate how to graph the parabola from the general form, including the focus and directrix.
Unit 6 Lesson 6 - In this video I demonstrate how to determine if a polynomial in general form is a hyperbola. I then use completing the square to convert it to the standard form for a hyperbola. Finally, I demonstrate how to graph it and find its foci.
Unit 7 Lesson 1 - In this video I introduce the concept of a sequence with its notation. I then explain the difference between explicit and recursive sequence definitions, working out the first 5 terms of each.
Unit 7 Lesson 2 - In this video I demonstrate how to write the explicit formula and the recursive formulas for any given arithmetic sequence.
Unit 7 Lesson 3 - In this video I review what an arithmetic sequence is, and define a sequence both recursively and explicitly. I then demonstrate how to find the first term given the common difference and a specified term, how to find any other term asked for, and how to find missing terms between two given terms in an arithmetic sequence.
Unit 7 Lesson 4 - In this video I explain the difference between a sequence and a series. I then use a specific example to derive the formula for the partial sum of an arithmetic series.
Unit 7 Lesson 5 - In this video I introduce sigma notation for the partial sum of series in general. I then demonstrate how to write an arithmetic series in sigma notation. I present the various known summation formulas for k, i, i^2, and i^3 and use the properties of summation to evaluate a given finite series.
Unit 7 Lesson 6 - In this video I review the recursive and explicit definitions of an arithmetic sequence. I then present an example of a geometric sequence and define it recursively and explicitly. Finally, I demonstrate how to find a previous term given the nth term and the common ratio, and how to find missing terms between two given terms.
Unit 7 Lesson 7 - In this video I demonstrate how to find the first term of a geometric sequence given two terms, as well as finding any subsequent terms. I then demonstrate how to find missing terms between any two given terms.
Unit 7 Lesson 8 - In this video I explain how it is that adding an infinite number of terms can result in a fine sum or total. I then give the harmonic series as an example of a divergent series and that a geometric series in which |r| < 1 is always convergent, having a finite sum.
Unit 7 Lesson 9 - In this video I review what a geometric series is. I then present the idea of a finite sum of a geometric series. I then derive the formula for the sum of a finite geometric series and use it to find the sum of the first 100 terms of the series 5 + 10 + 20 + 40 + 80 + ... .
Unit 7 Lesson 10 - In this video I derive the formula for an infinite geometric series from that for a finite series. I then use the formula to find the sum of an infinite series.