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How do write the exponential equation $2^x=8$ in logarithmic form? How do you write the logarithmic equation $\log_2 8 = x$ in exponential form? In case you didn’t notice, these two statements are mathematically equivalent, since the solution to both equals $3$, but what just happened? In the exponential equation,…
Since logarithmic functions are the inverse of an exponential functions, they must follow the laws of exponents. The three properties of logs listed below will be used to combine and solve expressions and equations in this section and the next. Each of these properties are derived from the laws of exponents.…
A radical equation is one in which the unknown variable (e.g. $x$) is under the radical sign. For example: $$\sqrt{x+5}=10$$ When it comes to solving radical equations, oftentimes your teacher will expect you to check your answer using the original equation. This involves substituting your answer back into the equation…
Add and Subtract Radicals One reason to learn how to simplify radicals is to be able to combine them through addition and subtraction. Radicals are called similar if they have the same index and same radicand, such as $4\cdot\sqrt[3]{xy}$ and $7\cdot\sqrt[3]{xy}$. Taking the sum of these two terms would result…
A radical is mathematical way to represent fractional exponents. A radical consists of a radical sign, a quantity under the radical sign called the radicand, and the index of the radical. If an expression, let’s say a, is raised to the power of ½, that’s the same as taking the square…
Radicals (√, ∛, ∜, etc.) are an extension of the exponents laws you learned in Part 1 of this course. This section is solely dedicated to the exponent laws. The connection between radicals and exponents is made in the next section, though it’s highly advised that you review these first as they’re easily…
Multiplying Complex Numbers Imaginary and complex numbers are multiplied the same way you multiply polynomials, with the addition of what you learned in the previous lesson about i when raised to varying exponents. Examples are shown below: Similarly, this idea can be expanded to imaginary numbers found within larger algebraic…
All the complex numbers you’ve dealt with this unit have been in rectangular form (a + bi). Polar form complex numbers are written in terms of r (radius length) and an angle (θ). The numbers a and b are related to r and θ by the formulas below: null In the first video…
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…