Algebra: Piecewise Function Review

Greetings, today’s post is for those in need of a piecewise function review!  This will cover how to graph each part of that oh so intimidating piecewise function.  There’s x’s, there are commas, there are inequalities, oh my! We’ll figure out what’s going on here and graph each part of the piecewise-function one step at a time.  Then check yourself with the practice questions at the end of this post. Happy calculating! 🙂

 

Wait, what are Piece-Wise Functions? Exactly what they sound like! A function that has multiple pieces or parts of a function.  Notice our function below has different pieces/parts to it.  There are different lines within, each with their own domain.Screen Shot 2020-07-21 at 10.01.59 AM

Now let’s look again at how to solve our example, solving step by step:

Screen Shot 2020-07-21 at 10.02.29 AM.pngScreen Shot 2020-07-21 at 10.02.41 AMScreen Shot 2020-07-21 at 10.03.06 AM.png

Translation: We are going to graph the line f(x)=x+1 for the domain where x > 0

To make sure all our x-values are greater than or equal to zero, we create a table plugging in x-values greater than or equal to zero into the first part of our function, x+1.  Then plot the coordinate points x and y on our graph.

Screen Shot 2020-07-21 at 10.04.33 AM

Screen Shot 2020-07-21 at 10.05.00 AM.png

 

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Translation: We are going to graph the line  f(x)=x-3 for the domain where x < 0

To make sure all our x-values are less than zero, let’s create a table plugging in negative x-values values leading up to zero into the second part of our function, x-3.  Then plot the coordinate points x and y on our graph.

Screen Shot 2020-07-21 at 10.07.33 AM.png

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Ready to try the practice problems below on your own!?

Practice Questions: Graph each piecewise function:

Screen Shot 2020-07-21 at 10.08.32 AM.png

 

 

 

 

 

 

 

 

Solutions:

Screen Shot 2020-07-21 at 10.09.20 AM

Screen Shot 2020-07-21 at 10.09.58 AM.png

Still got questions?  No problem! Check out the video above or comment below for any questions. Happy calculating! 🙂

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***Bonus! Want to test yourself with a similar NYS Regents question on piecewise functions?  Click here.

 

 

Geometry: Intersecting Secant Theorem

Ahoy! Today we’re going to cover the Intersecting Secants Theorem!  If you forgot what a secant is in the first place, don’t worry because all it is a line that goes through a circle.  Not so scary right? I was never scared of lines that go through circles before, no reason to start now.

If you have any questions about anything here, don’t hesitate to comment below and check out my video for more of an explanation. Stay positive math peeps and happy calculating! 🙂

Wait, what are Secants?

Screen Shot 2020-07-14 at 10.07.54 PM

Intersecting Secants Theorem: When secants intersect an amazing thing happens! Their line segments are in proportion, meaning we can use something called the Intersecting Secants Theorem to find missing line segments.  Check it out below: 

Screen Shot 2020-07-14 at 10.44.53 PM

Let’s now see how we can apply the intersecting Secants Theorem to find missing length.

Screen Shot 2020-07-14 at 10.45.29 PM.png

Screen Shot 2020-07-14 at 10.10.23 PMScreen Shot 2020-07-14 at 10.10.39 PM.pngScreen Shot 2020-07-14 at 10.11.13 PMScreen Shot 2020-07-14 at 10.11.52 PM.pngScreen Shot 2020-07-14 at 10.13.24 PMScreen Shot 2020-07-14 at 10.13.57 PM.pngScreen Shot 2020-07-14 at 10.14.20 PM

Screen Shot 2020-07-14 at 10.14.41 PM.png

Ready to try the practice problems below on your own!?

Practice Questions: Find the value of the missing line segments x.

Screen Shot 2020-07-14 at 10.38.02 PM

Screen Shot 2020-07-20 at 9.30.01 AM

Solutions:

Screen Shot 2020-07-20 at 9.30.55 AM.png

Still got questions?  No problem! Check out the video above or comment below for any questions and follow for the latest MathSux posts. Happy calculating! 🙂

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To review a similar NYS Regents question check out this post here

Algebra: 4 Ways to Factor Quadratic Equations

*If you haven’t done so, check out the video that goes over this exact problem, and don’t forget to subscribe!

Screen Shot 2020-06-02 at 3.03.55 PMScreen Shot 2020-06-02 at 3.04.24 PM____________________________________________________________________Screen Shot 2020-06-02 at 3.20.04 PM____________________________________________________________________

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Choose the factoring method that works best for you and try the practice problems on your own below!

Practice Questions:

Screen Shot 2020-06-02 at 3.09.58 PMSolutions:

Screen Shot 2020-06-02 at 3.10.30 PM

Want a review of all the different factoring methods out there?  Check out the ones left out here (DOTS and GCF) and happy calculating! 🙂

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Geometry: Median of a Trapezoid Theorem

*If you haven’t done so, check out the video that goes over this exact problem, also please don’t forget to subscribe!

Medians of a Trapezoid copy

Screen Shot 2020-06-02 at 7.31.07 AMStep 1:  Let’s apply the Median of a Trapezoid Theorem to this question!  A little rusty?  No problem, check out the Theorem below.

Median of a Trapezoid Theorem: The median of a trapezoid is equal to the sum of both bases.Screen Shot 2020-06-02 at 7.32.31 AMStep 2: Now that we found the value of x , we can plug it back into the equation for Screen Shot 2020-06-02 at 7.33.44 AMmedian,  to find the value of median Screen Shot 2020-06-02 at 7.34.25 AM

Screen Shot 2020-06-02 at 7.34.48 AM

Want more practice?  Your wish is my command! Check out the practice problems below:

Practice Questions:

1.Screen Shot 2020-06-02 at 7.35.29 AMis the median of trapezoid ABCDEF, find the value of the median, given the following:Screen Shot 2020-06-02 at 7.35.47 AM2. Screen Shot 2020-06-02 at 9.01.08 AMis the median of trapezoid ACTIVE, find the value of the median, given the following:Screen Shot 2020-06-02 at 9.16.22 AM3.Screen Shot 2020-06-02 at 9.17.01 AMis the median of  trapezoid DRAGON, find the value of the median, given the following:Screen Shot 2020-06-02 at 9.22.13 AM

4. Screen Shot 2020-06-02 at 9.23.08 AMis the median of trapezoid MATRIX, find the value of the median, given the following:Screen Shot 2020-06-02 at 9.23.43 AM

Solutions:

Screen Shot 2020-06-02 at 9.25.05 AM

Need more of an explanation?  Check out the detailed video and practice problems. Happy calculating! 🙂

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Algebra 2: Solving Radical Equations

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Today we’re back with Algebra 2, this time solving for radical equations!  Did you say “Radical Equations?” As in wild and crazy equations? No, not exactly, radicals in math are used to take the square root, cubed root, or whatever root of a number.

Radicals are actually pretty cool because we can write them a couple of different ways and they all mean the same thing! Check it out below:Screen Shot 2020-05-12 at 11.23.20 AM.pngStill not sure of their coolness? Let’s see what they look like with actual numbers:
Screen Shot 2020-05-12 at 11.24.21 AM.pngExample: Solve the following algebraic equation below for the missing variable (aka, solve for x).Screen Shot 2020-05-12 at 11.25.03 AM.pngExplanation:

How do I answer this question?   

The question wants us to solve for x using our knowledge of radicals and algebra. You can also check out how to solve this question on Youtube here!

How do we do this? 

Step 1: We start solving this radical equation like any other algebraic problem: by getting x alone. We can do this easily by subtracting 7 and then dividing out 5.Screen Shot 2020-05-12 at 11.26.21 AM.pngStep 2: Now, to get rid of that pesky radical, we need to square the entire radical.  Remember, whatever we do to one side of the equation, we must also do to the other side of the equation, therefore, we also square 14 on the other side of the equal sign. 

*This gets rid of our radical and allows us to solve for x algebraically as normal!Screen Shot 2020-05-12 at 11.29.11 AM.pngScreen Shot 2020-05-12 at 11.29.34 AM.pngWhat happens when there is a cubed root though!?!? When dividing polynomials with different value roots, raise the entire radical to that same power of root to cancel it out:Screen Shot 2020-05-12 at 11.30.21 AM.pngRemember, we know radicals can also be written as fractions: Screen Shot 2020-05-12 at 11.31.01 AM.pngTherefore we also know that if we raise the entire radical expression to the same power of the root, the two exponents will cancel each other out:Screen Shot 2020-05-12 at 11.31.47 AM.pngWant more practice? Try solving these next few examples on your own. Screen Shot 2020-05-12 at 11.32.39 AM.pngWhen you’re ready, check out the below:Screen Shot 2020-05-12 at 11.33.12 AM.png

Did I miss anything?  Don’t let any questions go unchecked and let me know in the comments! Happy calculating! 🙂 

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Algebra 2: Dividing Polynomials

Screen Shot 2020-05-03 at 11.43.01 AMNow that everyone is home, there is no better time to go over dividing polynomials! Whether school is out or not, dividing polynomials will always come in handy… I think.

Either way at some point, you may need to know how to answer these types of questions. The cool thing about dividing polynomials is that it’s the same long division you did way back in grade school (except now with a lot of x). Ok, let’s get to it and check out the question below:

Also, if you haven’t done so, check out the video related that corresponds to this problem on Youtube! 🙂

Screen Shot 2020-05-04 at 10.21.17 PMExplanation:

How do I answer this question?

The question wants us to divide polynomials by using long division.

How do we do this?     

Step 1: First we set up a good ole’ division problem with the divisor, dividend, and quotient to solve.Screen Shot 2020-05-04 at 10.43.48 PM.pngStep 2: Now we use long division like we used to back in the day! If you have any confusion about this please check out the video in this post.Screen Shot 2020-05-04 at 10.22.52 PMScreen Shot 2020-05-04 at 10.23.27 PM.pngWhat happens when there is a remainder though!?!? When dividing polynomials with a remainder in the quotient, the answer is found and checked in a very similar way! Check it out below:
Screen Shot 2020-05-04 at 10.23.44 PM

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Notice we represented the remainder by adding Screen Shot 2020-05-04 at 10.35.06 PM to our quotient! We just put the remainder over the divisor to represent this extra bit of solution.

Want more practice? Try solving these next few examples on your own. Screen Shot 2020-05-04 at 10.45.10 PM.pngWhen you’re ready, check out the solutions below:Screen Shot 2020-05-04 at 10.45.37 PMI hope everyone is finding something fun to do with all this extra time home! That can include everything from baking a cake to studying more math of course, happy calculating! 🙂 

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Earth Day Fractals!

In honor of Earth Day last week, I thought we’d take a look at some math that appears magically in nature.  Math? In nature?  For those of you who think math is unnatural or just terrible in general, this is a great time to be proven otherwise!

The key that links math to nature is all about PATTERNS. All math is based on is patterns.  This includes all types of math, from sequences to finding x, each mathematical procedure follows some type of pattern. Meanwhile back in the nearest forest, patterns are occurring everywhere in nature.

The rock star of all patterns would have to be FRACTALS. A Fractal is a repeating pattern that is ongoing and has different sizes of the exact same thing!  And the amazing thing is that we can actually find fractals in our neighbor’s local garden.

Let’s look at some Fractal Examples:

(1) Romanesco Broccoli:  Check out those repeating shapes, that have the same repeating shapes on those shapes and the same repeating shapes on even smaller shapes and…. my brain hurts!

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(2) Fern Leaves:  The largest part of this fractal is the entire fern leaf itself.  The next smaller and identical part is each individual “leaf” along the stem.  If you look closely, the pattern continues!

Screen Shot 2020-04-26 at 10.30.02 PM                                                        Screen Shot 2020-04-30 at 10.16.13 PM

(3) Leaves:  If you’ve ever gotten up real close to any type of leaf, you may have noticed the forever repeating pattern that gets smaller and smaller. Behold the power and fractal pattern of this mighty leaf below!

Screen Shot 2020-04-27 at 3.45.36 PM.                                                        Screen Shot 2020-04-30 at 10.16.55 PM

Just in case fractals are still a bit hard to grasp, check out the most famous Fractal below,  otherwise known as Sierpinski’s Triangle.  This example might not be found in your local back yard, but it’s the best way to see what a fractal truly is up close and infinite and stuff.

sierpinski.gif Screen Shot 2020-04-30 at 10.19.21 PM

Looking for more math in nature?  Check out this post on the Golden Ratio and happy calculating! 🙂

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