Let's look at finding the inverse of y=log(x) algebraically. First, recall that y=log(x) is shorthand for y=log10(x). Step 1: Swap x and y. Now we have x=log10(y). Step 2: Transform the log equation into exponential form. 10x=y Step 3: Replace y with the inverse notation of f-1(x). Thus, f-1(x)=10x Try a more challenging one on your own, such as f(x)=log2(x+3)
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Inverse Functions: Part 1
Algebra 2 typically involves the study of inverse functions, among other topics. If you take a function and reflect it over the line y=x, you have its inverse. Let's take y= log(x): Now, let's add the line y=x and the inverse of y=log(x), which is y=10x. We can now see the reflection over y=x: To graph inverse functions you may simply take all the points (x,y) of the original function and swap the x coordinates for the y coordinates. For y=log(x), you can see the first two coordina...
MoreFree Precalculus Textbook
Textbooks can be expensive. Fortunately, some professors are looking out for students' financial interests and have developed free and open textbooks. A collection of some of the current available books can be found at Stitz Zeager Open Source Mathematics. Professor's Stitz and Zeager offer a precalculus book released in July of 2013 that can be obtained in a pdf format. Here is a preview of the first two chapters:
MoreSeattle Math Tutor
Finding a Seattle math tutor is no easy chore. What do I look for in a tutor? Are you looking for a genius with a long resume of achievements in mathematics who can coach your child through math contests? Or does the tutor need to have expertise in special needs? As someone who hired a tutor while studying calculus at the University of Washington, I can relate. A brief phone call with a tutor is a great start to get a feel for what the tutor can offer. Meeting in person for a cup of coffe...
MoreSpeed and Distance
In Algebra 2, we may be asked to look at rates. Often these come in the form of speed and distance questions. The general formula for this type of problem is Continue Reading Speed and Distance
MoreSolving rational expressions
Sometimes you are presented with solving for unknown variables in rational expressions (expression where numerator and or denominator are polynomials).Continue Reading Solving rational expressions
MoreLogarithms: what are they?
A not so helpful definition of logarithms from 1797 Britannica: Logarithms are another way to consider unknown exponents. They ask: To what exponent do we raise this base to get another number?
Here are a few examples and explanations:
Log2(8) = x We interpret this as asking: “What number do we raise 2 to, to get 8”. So, let’s convert this into exponential form: 2x=8. We know that 2*2*2=8 so 23=8. Thus, x=3. If a problem simply has log without a base, it is assumed that the base is 10...
MoreFactoring by grouping
Most students find factoring fairly straight forward when first introduced. An equation might look like x2+6x+9 and be factored to (x+3)(x+3). But what happens when we see a coefficient to x2 that is not 1? This is where factoring by grouping comes into play. Here's an example: 4x2+16x-9 First, we need to find factors of the product of the a term (4) and the c term (-9) that add up to the b term (16). So first multiply 4*-9. We get -36. Now, let's look at all factors of -36 and find two th...
MoreTrigonometric ratios
When students are first exposed to trigonometry (the measurement of triangles) they are asked to memorize some trigonometric ratios. Here are a few of them: sin=opp/hypotenuse, cos=adjacent/hypotenuse, tan=opp/adjacent. Memorizing these ratios is important and can be aided by the following mnemonic device: sohcahtoa. It turns out one sailor in Seattle hasn't forgotten these ratios...in fact, he or she has opted to name their sailboat after this mnemonic device. I snapped this photo at a boaty...
MoreWhat are radians?
Angle measurement
The word trigonometry comes from Greek roots that loosely translate to the "measure of triangles." Most folks are familiar with using degrees to measures. Measurements of angles can be measured in degrees or radians. Converting between the two units of measurement is analogous to converting between other forms of measurement units. Radians reflect the length of an angle's corresponding arc created by the unit circle. If we recall the unit circle has a...
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