For Unit 6 – Rational Expressions and Equations, we learn a series of ideas such as: adding, subtracting, multiplying, and dividing rational expressions, solving for equations and more. In this blogpost, I will be going over 6.3 – Multiplying and Dividing Rational Expression. We were put to the task of creating a digital project that highlights our understanding of what we believe was the most important part of this unit. I choose to create a blog post describing all I know about this lesson.

Multiplying:

• When multiplying rational expressions (fractions)  its important you remember that just like multiplying regularly, you can just multiple directly across.
• However, before you can do so, we have follow a set os steps:

Example:

1. Step 1, Before you can begin multiplying you want to begin by checking all rational expressions (numerator and denominator) are in simplified form.
2. Step 2, you can now write your quotient as one fraction. Write it as a product of the factors of the numerators over the product of the factors of the denominators. Do not multiply yet.
3. Step 3, now we are able to cancel out common terms that are on the numerator and denominator. Simplify by dividing out the common factors (y + 3), (Y-3) and y.
   1. Note that even though all of the factors in the numerator were divided out there is still a 1 in there.  It is easy to think there there is “nothing” left and make the numerator disappear.  But when you divide a factor by itself there is actually a 1 there. 
4. Step 4, now you are able to re-write the simplified version.
   1. The values that would be excluded from the domain are 0, 3, -6, and -3.  Those are the values that makes the original denominator equal to 0. 0 cannot be the value of the denominator.

Dividing:

• Luckily,  diving rational expressions (fractions) the exact same as multiply with one more step prior. Its important you remember that just like dividing, you have to reciprocate and change the sign to multiplying.
• However, before you can do so, we have follow a set os steps:

Example:

1. Step 1, we begin by reciprocating the second quotient.
2. Step 2, Before you can begin multiplying you want to begin by checking all rational expressions (numerator and denominator) are in simplified form.
3. Step 3, you can now write your quotient as one fraction. Write it as a product of the factors of the numerators over the product of the factors of the denominators. Do not multiply yet.
4. Step 4, now we are able to cancel out common terms that are on the numerator and denominator. Simplify by dividing out the common factors 3x and (X+6)
   1. Note that even though all of the factors in the numerator were divided out there is still a 1 in there.  It is easy to think there there is “nothing” left and make the numerator disappear.  But when you divide a factor by itself there is actually a 1 there.
5. Step 5, now you are able to re-write the simplified version.
   1. The values that would be excluded from the domain are -6 and 0.  Those are the values that makes the original denominator of the product equal to 0.

For our Chapter 4 Summary Project, we were put to the task of creating a Parabola using the numbers of our birthday. My Example – October 21, 2004, would become  Then describe/explain everything you know about this as well as share a picture of what the graph looks like!

There are three different forms that you can write a quadratic function: Vertex form which is , Standard form which is  the last one is Factored form which is .

When graphing your parabola, it is important to know that each form of the function offers different pieces of information. In vertex form, you are given the translation (slides), stretch or compression, vertex, and line of symmetry. In Standard, you are given the Y-intercept. In the Factored form, you are given the X-intercepts. Transferring from form to form required you to do a series of completing the square, factoring, or expanding. The plus of graphing is that you can easily define each part of the parabola, by just analyzing your function.

• Vertex to Standard: Expand (FOIL/Distribute)
• Vertex to Factored: Must go through the standard first
• Standard to Vertex: Completing the square
• Standard to Factored: Factoring
• Factored to Vertex: Must go through standard first
• Factored to Standard: Expand (FOIL/Distribute)

Vertex Form:

• My equation in vertex form is
• The benefit of using vertex form is that it actually gives you a lot of information: translation (slides), stretch or compression, vertex, and line of symmetry.

• For my example
• We can easily identify the stretch (the coefficient of the brackets, “a”): -3
  • First things first, we can identify that the parabola opens downward or reflects as it is negative.
  • Because the number is larger than one, the parabola will be narrower. This is because all the values of the parent function will be multiplied 3x larger.
• Next, we can identify the horizontal translations. The Horizontal translation is what is in the brackets. If the number is -, it translates to move to the right a number of spaces, and if it’s +, it translates to move to the left a number of spaces.
  • In my parabola, it is (x – 3), therefore our slide moves 3 units to the right.
• Then, the last number is constant (a number that has nothing changing it). That number will determine our vertical slide.
  • In my equation, the constant is  + 4. That means we slide the parabola 4 units up vertically.
• Finding the Vertex
  • The good thing about finding the vertex from vertex form is that you can simply look at your equation and determine the X and Y values.
  • In the image above, our X coordinate (aka h) and our Y coordinate (aka K) can just be substituted.
  • The vertex for my parabola is: (3, 4) *remembering that our horizontal slide moved to the right (positive)

Standard Form:

• My equation in standard form is
• The benefit of using the standard form is that it gives you the Y-intercept and the stretch/compression.

• After completing your steps to get into standard form (expanding), you can identify the y-intercept by finding the constant. As shown in the image above, the constant (-7) is the Y-intercept.
  • In my parabola, the Y-intercept is -23

Factored form:

• The benefit of using the factored form is that it actually gives your X-intercepts and the stretch/compression.
• For my example, the standard form does not nicely factor, and the X-intercepts are not whole numbers. They are decimal points and can be determined as mixed radicals/fractions using the quadratic formula.
• X-intercepts =  (1.845, 0) and (4.155, 0)

My Parabola:

Description:

Vertex = (3,4)

X-intercepts =  (1.845, 0) and (4.155, 0)

Y-intercepts = (0, -23)

Stretch = -3

Opens = Down (reflections)

Horizontal translation = 3 units to the right

Vertical Translation = 4 units up

For Unit 3 – Solving Quadratic Equations, we learn a series of ideas such as: factoring binomials and binomials, completing the square and more. I believe the most important lesson was 3.5 – Solving quadratic equations using the Quadratic Formula. We were put to the task of creating a digital project that highlights our understanding of what we believe was the most important part of this unit. I choose to make a digital poster as enjoy creating them and am familiar with using Glogster.

*Zoom in if necessary

For Unit 2 – Radical Operation and Equations, we learn a series of ideas such as adding and subtracting radicals expressions, multiplying and dividing radical expressions, solving radicals algebraically and more. I believe the most important lesson was 2.3 Multiplying and Dividing Radical Expressions, specifically dealing with irrational denominators. We were put to the task of creating a digital project that highlights our understanding of what we believe was the most important part of this unit. I chose to make a video.

Core-Competency:

For Unit 1 – Roots and Powers, we learn a series of ideas such as, classifying real numbers, writing mixed/entire radicals, evaluating radicals with positive or negative rational exponents, and more. I believe the most important lesson was 1.5 Evaluating powers with rational exponents. We were put to the task of creating a digital project that highlights our understanding of what we believe was the most important part of this unit. I choose to make a digital poster as enjoy creating them and am familiar with using Glogster.