Unit 7 (March)
Simplifying radicals (Radical Properties): Recall from Chapter 6 your exponent rules. Now recall that an exponent (specifically a fractional exponent) can be expressed using exponents or a radical symbol. For example:
To simplify radicals, we rely on adapting our existing exponent rules. For example, if:
By using our knowledge of exponents, we now know that the square root of any number, times itself, is equal to that same number. For example, the square root of 7 times the square root of 7 equals 7.
Rather than show you how we create each rule (that's what class is for), the radical rules you are responsible for knowing are as follows:
Rather than show you how we create each rule (that's what class is for), the radical rules you are responsible for knowing are as follows:
The rules for knowing when a radical is "fully simplified" are as follows:
Video examples of simplifying radicals are below the example below
Simplifying using Prime Factorization: Recall from your younger years that 'prime factorization' means to break up a number into a product of its prime factors. For example, the PF of 36 is 2x2x3x3 or 2^2 x 3^2. We can use this basic concept in conjunction with our radicals rules to simplify radicals because we know that the m root of any number to the m is itself. What??? For example, the 5th root of 21^5 is 21. The square root of 10^2 is 10. Think about that second one....the square root of 10^2 (or 100) is 10. See how this plays out below:
- There are no radicals in the denominator of any fraction. If there are, you must "rationalize" your denominator
- No factors can be 'pulled out' or cancelled out of your radicals
Video examples of simplifying radicals are below the example below
Simplifying using Prime Factorization: Recall from your younger years that 'prime factorization' means to break up a number into a product of its prime factors. For example, the PF of 36 is 2x2x3x3 or 2^2 x 3^2. We can use this basic concept in conjunction with our radicals rules to simplify radicals because we know that the m root of any number to the m is itself. What??? For example, the 5th root of 21^5 is 21. The square root of 10^2 is 10. Think about that second one....the square root of 10^2 (or 100) is 10. See how this plays out below:
Video examples (scroll through the left side to see tons of examples besides the one given)
Solving quadratic equations algebraically: Using our newfound knowledge of radicals, we can solve quadratic equations algebraically. We now know that to 'cancel' out a square, we square root the expression, to cancel out a cube, we cube root the expression, and so on.....
Notice how the answer is not simply 3, it is 3 and -3. That is because the square root of 9 is not simply 3, -3 times itself is also equal to 9. To visualize why this happens, think about how you would go about solving the equation above graphically....
By graphing this quadratic, we see that there are 2 unique roots or x-intercepts. One at -3 and one at +3. Therefore there are algebraically 2 solutions that would make this quadratic work.
Imaginary numbers: Imaginary numbers are numbers that , when squared, return a negative value. In other words, the square root of -9 is an imaginary number. We spend a good deal of class time explaining this concept, therefore, the explanations below are more for the 'algebraic' or mathematical uses of imaginary numbers.
Imaginary numbers: Imaginary numbers are numbers that , when squared, return a negative value. In other words, the square root of -9 is an imaginary number. We spend a good deal of class time explaining this concept, therefore, the explanations below are more for the 'algebraic' or mathematical uses of imaginary numbers.
The i cycle: Since we know that i = sqrt(-1), we can readily find values of i^2, i^3, i^4 and so on simply by multiplying i. What we notice is a pattern, or cycle, where every 4th exponent (i^4, i^8, i^12, etc...) is equal to 1. Therefore, the next value in the pattern must be i itself. See the table that we created in class below:
Simplifying Imaginary numbers: Now that we have a way to write imaginary numbers, we also have a way to simplify radicals involving negatives. See the examples below for proper simplification:
One thing to note regarding simplifying with imaginary numbers: You must first deal with the imaginary parts first before you simplify. For example, using our multiplication property of radicals the example on the right could have been turned into sqrt(36), which would simplify into 6. However, we would be inadvertently getting rid of the imaginary part of each radical, and therefore 'pretending' that the original numbers weren't imaginary at all. Therefore, we first had to take out the imaginary number i, then simplify the radicals using our rules.
Operations with Imaginary numbers: Since we know how to perform operations with radicals and exponents, we are now able to perform operations with imaginary numbers too (the rules are all the same after all). See the examples below.
Solving equations that have Imaginary solutions (non-real solutions): We are also able to solve quadratics equations that previously had "no real number solution":
Completing the Square: CTS is one of the cornerstones of Algebra. It can be tricky to comprehend at first, but it is one of those concepts that reappears most often in your CAPT tests, SATs, College placements, etc etc.... In completing the square, the easiest way to imagine it is to think back to when we first started factoring, and what it would look like if we could not create a specific shape with our Algebra tiles. This would mean that we could not factor the equation. Well in 'completing the square', we are imagining what we would have to add or subtract to the equation if we were to create a perfect square. Visually....
In completing the square, you use this method to actually solve the quadratic equation because we know that if we can create a perfect square, that means we can then factor that shape. Algebraically speaking:
Some rules for completing the square are as follows:
1. You can complete the square for ANY quadratic. This is a foolproof method that will always work, however, some methods (like factoring) may be easier.
2. Be aware of your radical rules!!! For example, you cannot take the square root of a negative number, so if at any point you find yourself square rooting a negative, you know you cannot proceed any further with real numbers. This quadratic has no real solution.
3. To find the value you add on in step 1, you take half of your b term and square it.
Video examples of completing the square
Solving quadratics with the quadratic formula: The quadratic formula is created with the knowledge that you can complete the square on any quadratic. Since you can solve a quadratic with CTW, that stands to reason that you must also be able to solve the standard form of a quadratic equation. While it isn't pretty, this is also a 'corner stone' of Algebra. The derivation or creation of the Quadratic Formula is a method seen in all levels of math, and is often revisited in Algebra 2, pre calculus, CAPT, SAT, etc... You will therefore also be asked to do this on your next quiz :)
It stands to reason that if you can solve quadratic with Completing the Square, then you must be able to solve the standard form of a quadratic (ax^2 + bx + c=0) for x. This would give us a formula that we could plug any quadratic into and have it automatically give us the solutions.
See the video below on deriving the quadratic formula:
1. You can complete the square for ANY quadratic. This is a foolproof method that will always work, however, some methods (like factoring) may be easier.
2. Be aware of your radical rules!!! For example, you cannot take the square root of a negative number, so if at any point you find yourself square rooting a negative, you know you cannot proceed any further with real numbers. This quadratic has no real solution.
3. To find the value you add on in step 1, you take half of your b term and square it.
Video examples of completing the square
Solving quadratics with the quadratic formula: The quadratic formula is created with the knowledge that you can complete the square on any quadratic. Since you can solve a quadratic with CTW, that stands to reason that you must also be able to solve the standard form of a quadratic equation. While it isn't pretty, this is also a 'corner stone' of Algebra. The derivation or creation of the Quadratic Formula is a method seen in all levels of math, and is often revisited in Algebra 2, pre calculus, CAPT, SAT, etc... You will therefore also be asked to do this on your next quiz :)
It stands to reason that if you can solve quadratic with Completing the Square, then you must be able to solve the standard form of a quadratic (ax^2 + bx + c=0) for x. This would give us a formula that we could plug any quadratic into and have it automatically give us the solutions.
See the video below on deriving the quadratic formula:
To make things easier to see all at once, the following work was taken off of purplemath.com:
This is the quadratic formula. All we need is to determine the values for a,b, and c in quadratic, and we can plug them into this formula to determine the solutions for x.
Thinking back to our parabola shape.... We typically have two roots, a vertex, and an axis of symmetry. If we take a look at the second to last step above, we see -b/2a plus or minus another fraction. Thinking logically, if we have a value (-in this case, -b/2a) and then add some number and subtract that same number, then -b/2a must be the middle value in between 2 equally spaced outside values. Hence, -b/2a is actually the 'formula' for finding the axis of symmetry, and the square root fraction that you add and subtract refers to the root to the right of the axis and the root to the left.
Note: You can only use the quadratic formula when your equation is set equal to zero
Examples of using the quadratic formula
Calculating and using the discriminant of the quadratic formula: The discriminant is the name used for the 'stuff' underneath the radical in the quadratic formula above (b^2-4ac). Why is this important? Think about it. If the expression underneath the radical is negative, what happens? You can't take the square root of a negative! So if you happen to plug in values for a, b, and c and get a negative value inside the radical, then you can go no further. This quadratic equation will have no solution. The quadratic equation itself can only be used when your original quadratic is equal to 0. So what does this mean? If you cannot successfully solve a quadratic because you get "no solution", then that means there is no value you can plug in for x that will make the quadratic equal to 0, or the parabola shape equal to 0 (it will not touch the x-axis). You will have no roots.
Likewise, what happens if the values of the discriminant (b^2-4ac) happen to equal 0? Then your numerator will look something like this: -b +/- sqrt(0). The square root of 0 is 0, so you are essentially adding and subtracting nothing, and getting the same solution when you apply the quadratic formula. Huh? Graphically speaking, if the roots of my parabola both lie on the same point on the x-axis, then I must only have one root.
Thinking back to our parabola shape.... We typically have two roots, a vertex, and an axis of symmetry. If we take a look at the second to last step above, we see -b/2a plus or minus another fraction. Thinking logically, if we have a value (-in this case, -b/2a) and then add some number and subtract that same number, then -b/2a must be the middle value in between 2 equally spaced outside values. Hence, -b/2a is actually the 'formula' for finding the axis of symmetry, and the square root fraction that you add and subtract refers to the root to the right of the axis and the root to the left.
Note: You can only use the quadratic formula when your equation is set equal to zero
Examples of using the quadratic formula
Calculating and using the discriminant of the quadratic formula: The discriminant is the name used for the 'stuff' underneath the radical in the quadratic formula above (b^2-4ac). Why is this important? Think about it. If the expression underneath the radical is negative, what happens? You can't take the square root of a negative! So if you happen to plug in values for a, b, and c and get a negative value inside the radical, then you can go no further. This quadratic equation will have no solution. The quadratic equation itself can only be used when your original quadratic is equal to 0. So what does this mean? If you cannot successfully solve a quadratic because you get "no solution", then that means there is no value you can plug in for x that will make the quadratic equal to 0, or the parabola shape equal to 0 (it will not touch the x-axis). You will have no roots.
Likewise, what happens if the values of the discriminant (b^2-4ac) happen to equal 0? Then your numerator will look something like this: -b +/- sqrt(0). The square root of 0 is 0, so you are essentially adding and subtracting nothing, and getting the same solution when you apply the quadratic formula. Huh? Graphically speaking, if the roots of my parabola both lie on the same point on the x-axis, then I must only have one root.
The above chart was taken from: Regents test prep
Unit 8 (April)
Graphing Square Roots and Functions: The graph of a square root function is similar to the of a parabola, except the complete opposite :) The standard form of a square root function is in the form:
Where the starting point is at (h,k) [recall how the vertex of a parabola is also at (h,k)], and a represents how fast the function increases and in which direction, up or down (similar to how a affects the vertex form of a quadratic). We must also find the x and y-intercepts (if any). However, those should be easy by now....x-intercept occurs when y = 0, y-intercept occurs when x=0.
Note: You can ALWAYS resort back to making a common x/y table, plugging in values for x, and getting values for y in return (points). This can be used with ANY equation, and is how we learn how to graph, but I am only providing the 'short cuts' here.
When graphing a square root function, the most important thing to note are the restrictions on the domain and range. Specifically, note how a square root acts when you place different numbers inside the radical (like 0 or negatives). For example, consider the equation y = sqrt(x - 2) - 1.
You cannot have a square root of a negative number, therefore x cannot be less than 2. An x-value less than 2 (like 1) would return a negative value inside the radical, and therefore cannot be graphed. Likewise, since 2 is the lowest x value, that means sqrt(2-2) - 1 (or 0 - 1, which is -1) is the lowest y value (or highest if your a value is negative). See the examples below:
Note: You can ALWAYS resort back to making a common x/y table, plugging in values for x, and getting values for y in return (points). This can be used with ANY equation, and is how we learn how to graph, but I am only providing the 'short cuts' here.
When graphing a square root function, the most important thing to note are the restrictions on the domain and range. Specifically, note how a square root acts when you place different numbers inside the radical (like 0 or negatives). For example, consider the equation y = sqrt(x - 2) - 1.
You cannot have a square root of a negative number, therefore x cannot be less than 2. An x-value less than 2 (like 1) would return a negative value inside the radical, and therefore cannot be graphed. Likewise, since 2 is the lowest x value, that means sqrt(2-2) - 1 (or 0 - 1, which is -1) is the lowest y value (or highest if your a value is negative). See the examples below:
Solving Square Root equations: Because by now you should have extensive knowledge of how squares and square roots behave, you should have little trouble on this concept. However, I have provided a few examples below. Be aware that you WILL get extraneous solutions often when solving square root equations. Two reasons for this: 1) You cannot have negative numbers inside the radical, and 2) You cannot have a square root equaling a negative number.
Graphing Cube Root functions: Cube root functions act very similar to square root functions. Think about it..... With a square root, you are finding out numbers that multiply together to form another number. Whereas with a cube root, you are finding numbers that multiply together 3 times.
Cube root functions, for our use in this class, are in the form:
Where the starting point is at (h,k) [recall how the starting point of a square root function is also at (h,k)], and a represents how fast the function increases and in which direction, up or down (similar to how a affects the vertex form of a quadratic). We must also find the x and y-intercepts (if any).
The difference between square root functions and cube root functions? You can have cube root of negative numbers (the cube root of -27 is -3 because -3 x -3 x -3 = =27, but you cannot have a square root of a negative number. What does this mean as far as the graph goes? It means that for a cube root function you can have negative values inside the radical, meaning the domain of the function is all real numbers. See example below:
The difference between square root functions and cube root functions? You can have cube root of negative numbers (the cube root of -27 is -3 because -3 x -3 x -3 = =27, but you cannot have a square root of a negative number. What does this mean as far as the graph goes? It means that for a cube root function you can have negative values inside the radical, meaning the domain of the function is all real numbers. See example below:
Inverse functions: Recall that when we learned direct variation (y = kx), we found the inverse variation by swapping the x and y, and solving for y (so x = yk, divide both sides by k, and you get y = k/x).
To get the inverse of an equation, simply swap the x and y values, and rewrite the equation in function form (y = form). Graphically, doing this is the transformation equivalent of reflecting over the y=x line. This explains the rule for reflecting over the y=x line, that (x,y) ---> (y,x) (oh yeah, remember we did that???).
Note: BE AWARE, any restrictions or limitations that exist on the original equation must translate to the new equations too. The best way to explain this if through the inverse of a square root function. For example:
To get the inverse of an equation, simply swap the x and y values, and rewrite the equation in function form (y = form). Graphically, doing this is the transformation equivalent of reflecting over the y=x line. This explains the rule for reflecting over the y=x line, that (x,y) ---> (y,x) (oh yeah, remember we did that???).
Note: BE AWARE, any restrictions or limitations that exist on the original equation must translate to the new equations too. The best way to explain this if through the inverse of a square root function. For example:
Notice the original function (in blue) appear to be reflected over the y=x line. Simple enough right? However, algebraically things get a tad tricky. Aside from actually showing the work to find the inverse, notice how the inverse is actually a type of quadratic. Shouldn't the inverse then be a parabola shape??? NO!!! If we take a look at the original function once more, notice that you cannot have an x value less than -4 (otherwise you would have a negative inside the radical). Therefore, you have a limitation on the original domain where x > -4, and on the range where y > 2.
On the new inverse equation, we flip the x and y, therefore, the limitations also flip. So on the inverse, y > -4 and x > 2. What does this mean? Although your inverse function is indeed a parabola, it has a limitation so you only end up graphing the section allowed. Your inverse function looks like this:
On the new inverse equation, we flip the x and y, therefore, the limitations also flip. So on the inverse, y > -4 and x > 2. What does this mean? Although your inverse function is indeed a parabola, it has a limitation so you only end up graphing the section allowed. Your inverse function looks like this:
To imagine things easier, you can do two things. First, think back to when we did piecewise functions. Essentially, the inverse here is a piecewise function. Second, look at the graph of the original equation. There is a clear starting point, and the inverse much reflect / match it perfectly. Regardless of the equation you find for the inverse, your restrictions must mathematically align with the domain / range of the original.
Note: Not all inverses will have restrictions, In fact, most do not. Only functions that originally have restrictions will have inverses that also have them. For example, think of y = 2x + 6. In this original equation x can be any number, therefore y can be any number in the inverse. Equations that will have certain types of restrictions are square root functions, absolute value functions, and rational functions.
Note: Not all inverses will have restrictions, In fact, most do not. Only functions that originally have restrictions will have inverses that also have them. For example, think of y = 2x + 6. In this original equation x can be any number, therefore y can be any number in the inverse. Equations that will have certain types of restrictions are square root functions, absolute value functions, and rational functions.