Problem Solved!
Up to this point, we have avoided taking the square root of negative numbers, such as √(−1). We’ve simply dismissed the notion by saying an output does not exist, and rightfully so given the real number system we’ve been brought up with. In this section, we deal with them by introducing…
This section caters exclusively to word problems involving combined variation. As introduced in the previous section, combined variation is when there is both a direct (multiplication) and inverse (division) variation that occurs together. Be sure to watch all three parts as they examine different scenarios commonly found in textbooks, all of…
When y varies directly as both x and w, we say that y varies jointly as x and w. When you first looked at direct variation, you focused mainly on a single dependent and independent variable (i.e. y and x, respectively). This time the equation directly depends one 2 or…
The phrase “y varies inversely as x” or “y is inversely proportional to x” means that as x gets bigger, y gets smaller, and vice versa. Inversely proportional terms can mathematically be represented as: null Notice how x is under a constant k. ? Don’t confuse inverse variation with inverse functions. An inverse function…
Direct variation is when the dependent variable, y, varies according to the independent variable, x. The most generic direct variation equation is y = 1·x. If you plot this equation, you will get a diagonal line cross the origin. To generalize any direction variation equation, use the following template: null…
In the first lesson of this unit, you were lightly introduced to graphing polar coordinates. This lesson revisits what you learned earlier, and extends those ideas to graphing polar equations. In the first of three videos below, you will learn how to graph polar coordinates with negative values, namely when:…
Just as in the previous section, you can convert rectangular equations to polar equations using the same formulas introduced before (summarized below). Rectangular equations are written exclusively in terms of x and y, while polar equations are written in terms of r and θ. The first of many videos related to this…
Up until now, you have worked purely with a rectangular coordinate system, in which there are two perpendicular axes, and points are specified according to their coordinates, (x, y). What’s interesting is that whenever you plot a point on an x-y plane, that point can also be represented as a vector,…
The final section of this chapter involves making an equation from a waveform. In other words, you’ll be shown a wave, and you’ll be expected to identify its amplitude, period, and phase-shift, then use this information to generate an equation using one of the templates shown below. Since a sine…
Oftentimes you’ll be presented with the properties of the wave and will be expected to create an equation from them. For example, you may be given the amplitude, period, phase-shift or told that it reflects about the x axis or has a height of n. Whatever the case is, you…
Now that you know how to identify the amplitude, phase shift, and cycle when given a periodic sinusoidal function, it’s time you learn how they’re graphed via the steps outlined underneath. Be mindful that these steps are identical for sinusoidal functions containing cosine, with one exception in Step D – the…
All cosine functions start off looking like this without applying any transformation: The steps to graphing cosine waves is identical to graphing sine waves. A summary of steps as explained in the previous video are written below: (a) Draw two horizontal lines, each at a distance equal to the amplitude…
In Part 1 of this course (Math 1131), you were newly introduced to the trigonometric functions: sine, cosine, and tangent. You learned how you can use these functions to solve triangles by setting up ratios, but you never learned what they looked like graphed. It turns out that if you…
Many applications in real-life contain two or more unknowns. To solve such problems, we must write as many independent equations as there are unknowns. Otherwise, it is not possible to obtain numerical answers. This section mirrors what you’ve already been doing this unit – that is, solving systems with two or…
There are practically countless methods devised over the past millennia that have enabled mathematicians to solve systems with three unknowns. In this section, we will focus on one method exclusively which is based on the methods that you’ve already learned when you solved systems with two unknowns. Technically, when a…
In this section, you’ll use the method of addition-subtraction or the method of substitution to solve systems of equations with literal coefficients (coefficients that aren’t numbers). To do this, you treat the coefficients as if they were numbers, keeping in mind what you’ve learned in the past about combining like-terms.…
This lesson will explore how to use the techniques used in the previous method to solve non-linear equations. Specifically, we’ll look at a system of equations whose variable is found in the denominator. null Notice that both equations displayed above consist of terms whose variables (x and y) are positioned…
Elimination Method In the elimination method (also known as the addition-subtraction method), we eliminate one of the unknowns by first (if necessary) multiplying each equation by a factor will make the coefficients of one unknown in both equations equal, regardless of their signs. The two equations are then added or subtracted…
This chapter introduces us to solving a system of equations. In part one of this course (Math 1131), you learned how to solve linear equations (such as, 25 = x + 5), for the unknown variables. This time you’ll learn to solve a system of linear equations with two unknowns,…
A numerical operation can be described as an action or process used to solved a numerical problem. Adding and Subtracting Signed Numbers If you have two numbers x and y, the following rules apply when these numbers are being added or subtracted. Rule of Signs for Addition: (x – y)…