*John Hamilton*Show bio

John has tutored algebra and SAT Prep and has a B.A. degree with a major in psychology and a minor in mathematics from Christopher Newport University.

Instructor:
*John Hamilton*
Show bio

John has tutored algebra and SAT Prep and has a B.A. degree with a major in psychology and a minor in mathematics from Christopher Newport University.

The ensuing year-end Algebra I homeschool assignment will challenge students on several math concepts, including how to factor and graph a quadratic equation. This mathematics assignment has been created for the 9th grade student.
Updated: 02/27/2021

When your students think of algebra, including **exponents**, **polynomials**, and **graphs**, the last thing they probably think about is **geometry**, right? Well, there do exist some subtle connections between the two disciplines.

In addition to what is mentioned above, this synoptic end-of-year assignment will cover topics learned throughout the two freshman Algebra I school semesters, including the distributive property along with the five laws of exponents too.

By the end of this assignment, your students will have completed four steps, solved three problems, and delivered two presentations.

Note - The answers are located at the bottom of the page.

- For even more math assistance, visit our High School Algebra I: Homeschool Curriculum.

**Algebra:**a branch of mathematics in which letters represent numbers**Exponent:**a quantity which represents a power to be raised**Geometry:**a branch of mathematics in which properties of figures are explained

- Graph paper
- Online capability
- Paper
- Pencil
- Ruler

- Two days to complete this Algebra I assignment
- Two weeks to design an appropriate presentation

Let's begin! Do you remember the five laws of exponents from earlier this school year? They are:

**Product of Powers Rule:**

*x**a***x**b*=*x**a*+*b*

Example:

- 25 * 26 = 2(5 + 6)

**Quotient of Powers Rule:**

*x**a*/*x**b*=*x**a*-*b*

Example:

- 38 / 35 = 3(8 - 5)
- Of course,
*x*doesn't equal zero, because you can't divide by zero.

- Of course,

**Power of a Power Property:**

- (
*x**a*)*b*=*x*(*a***b*)

Example:

- (42)3 = 4(2 * 3)

**Power of a Product:**

- (
*x***y*)*a*=*x**a***y**a*

Example:

- (5 * 6)4 = 54 * 64

**Power of a Quotient:**

- (
*x*/*y*)*a*=*x**a*/*y**a*

Example:

- (6 / 7)5 = 65 / 75

**Problem #1:**

Next, let's review what we learned when you want to factor the following quadratic equation:

*x*2 - 6*x*- 16

**Problem #2:**

Now do you recall how to graph this same quadratic equation?

*f*(*x*) =*x*2 - 6*x*- 16

Okay, you remember quadratic equations, but do you remember how to solve those ever-so-dastardly cubic equations?

- x3 - 4x2 + x + 6 = 0

This is a bit tricky, but you can handle it. First, we apply the **rational roots theorem** or **rational roots test**. This means all our possible solutions are found by dividing all the factors of the constant (which is 6) by the leading coefficient (the number in front of *x*3, which in this case is understood to be 1), and include positives and negatives in your list.

- Factors of 6 = 1, 2, 3, 6
- Factor of 1 = 1

So our possible solutions are +/- 1, +/-2, +/-3, and +/- 6.

Now I have some bad news, but you may remember we don't have a cool, convenient formula for cubic equations like we do for those quadratic equations. Yep! You actually have to plug in each of these eight possible solutions to find your three solutions. On the other hand, 3 out of 8 aren't horrible odds, right?

Let's start with 1:

- x3 - 4x2 + x + 6 = 0
- (1)3 - 4(1)2 + (1) + 6 = 0
- 1 - 4 + 1 + 6 = 0
- 4 = 0

Hey, it didn't work. Boo! Hiss! Hiss! Boo! Oh well, let's try again with -1.

- x3 - 4x2 + x + 6 = 0
- (-1)3 - 4(-1)2 +(-1) + 6 = 0
- -1 - 4 - 1 + 6 = 0
- -6 + 6 = 0
- 0 = 0

It worked! Since -1 is indeed a zero of the equation, we know (*x* + 1) is a factor of the equation. Now, to find the other two factors, go ahead and plug in the other 6 numbers. I'll give you 15 minutes.

Are you done?

You should know by this point the three roots, solutions, or zeros are:

- -1, 2, and 3

Or your three factors are:

- (
*x*+ 1) (*x*- 2) (*x*- 3)

By the way, let's recount some of the properties of polynomial functions. First of all, their graphs are continuous. In other words, when drawing one you never lift your pencil from your paper. Second, their graphs are smooth. In other words, your graph will have no sharp corners, but have rounded curves instead.

''How does measurement in algebra relate to geometry though? I thought we already learned algebra and geometry were two completely separate mathematical fields.'' Well, not completely. When you graphed your above polynomial function, you were essentially converting an algebraic concept into a geometric concept.

In a reverse manner, we might be given a graph of a geometric circle, but we use our handy algebra formulas to solve for the perimeter and the area of that circle.

And of course, the Pythagorean theorem combines algebra and geometry concepts.

Now let's review the concept of algebraic distribution. You may remember this formula:

*a*(*b*+*c*) =*a**b*+*a**c*

We usually read this expression as *a* times quantity *b* plus *c* equals *a* times *b* plus *a* times *c*.

**Problem #3:**

Distribute:

- 5(3
*x*+ 2*y*)

Now it's time to demonstrate your grasp of all the Algebra I material you learned throughout our school year by completing two of the following five presentations:

- Consider creative ways to represent graphs, such as using string or wool to design them.
- Create a year-end synoptic poster that answers questions related to all of the information covered in this assignment. Create one poster for the concepts you learned during each month, and then display each poster from left to right on a wall. You can take photos of the posters and post them online as well.
- Go ahead and ''be the teacher'' while walking through concepts and processes you learned sequentially this year, during your own ten-to-fifteen-minute YouTube video or TED Talk podcast.
- Make up your own year-end questions for a fun and interactive math quiz show.
- Write ten questions you can ask other students which assess understanding of the various concepts and processes.

**Solution #1:**

*x*2 - 6*x*- 16

First, ask yourself which numbers multiply to equal 16, and also add up to -6.

- (
*x*+ / - ?) (*x*+ / - ?)

Well, you know the factors of 16 are:

- 1 and 16
- 2 and 8
- 4 and 4

Since it's highly unlikely 1 and 16 add up to -6, let's choose 2 and 8 instead.

- (
*x*+ 8) (*x*- 2) = - (
*x*2 - 2*x*+ 8*x*- 16) = - (
*x*2 + 6*x*- 16)

It didn't work. ''I quit! This is just too difficult. I guess I'm going to go visit Taylor Swift's Facebook page instead.'' No, no. Don't quit. Give it one more attempt.

- (
*x*+ 2) (*x*- 8) = - (
*x*2 - 8*x*+ 2*x*- 16) = - (
*x*2 - 6*x*- 16)

It worked! Thank goodness we stayed focused and determined.

**Solution #2:**

*f*(*x*) =*x*2 - 6*x*- 16

Let's find our intercepts first:

- Set
*x*= 0 *f*'(0) = 0 - 0 - 16*f*(0) = -16

Now we know our *y*-intercept is (0, -16)

- Set
*y*= 0 - 0 =
*x*2 - 6*x*- 16 - 0 = (
*x*+ 2) (*x*- 8)

We already did that factoring in Part I, so:

*x*= -2 or*x*= 8

Therefore, our *x*-intercepts are (-2, 0) and (8, 0).

Now plot these three points on your graph paper. You know the graph of a quadratic is a parabola, so see if you can draw this one with just those three points. If not, go ahead and plug in a few more points before finishing the graph of your equation.

**Solution #3:**

- 5(3
*x*+ 2*y*) = - (15
*x*+ 10*y*)

Check your answer:

- (15
*x*+ 10*y*) =

Factor out a 5:

- 5(3
*x*+ 2*y*)

See how factoring is basically the opposite of distributing?

Requirements | 0-5 points |
---|---|

Problem #1 is answered correctly | |

Problem #2 is answered correctly | |

Problem #3 is answered correctly | |

Deliverable #1 is clear and concise | |

Deliverable #2 is clear and concise | |

Total |
/25 points |

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