# Precalculus Assignment - Quadratics, Functions & Graphing

Instructor: John Hamilton

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 following homeschool mathematics assignment deals with quadratic equations, including the three methods of completing the square, factoring, and the quadratic formula. The assignment has been designed for the 12th grade student. Updated: 04/06/2021

## Assignment Explanation and Topic Overview

Before moving on to calculus in college, your math students will need a solid foundation based in the knowledge of quadratic functions.

In addition to what we mentioned above, this lesson includes how to graph piecewise functions.

By the end of this assignment, students will have completed four steps, solved eight problems, and turned in two relevant presentations.

Note: The answers to the problems, including detailed explanations, are situated at the bottom of the page.

## Key Terms

• End behavior: how a given graph behaves for really small values as well as really large values
• Piecewise function: contains different parts or sub-functions
• Rational function: created by dividing a polynomial by a second polynomial

### Materials

• Graph paper
• Internet access
• Paper
• Pencil
• Ruler

### Time/ Length

• Three days to complete this precalculus assignment
• Two weeks to deliver two quality presentations

## Assignment Instructions for Students

### Step One

#### Part I

Are you ready for more fun than going to a cool water park on a hot summer day? Okay, not really, but math can be sort of fun as well. To start with today, let's learn how to solve quadratic equations by the three methodologies of completing the square, factoring, and utilizing the quadratic formula.

Problem # 1:

First, let's complete the square.

Solve for x2 - 6x + 11 = 0

Problem # 2:

Using factoring, solve the quadratic equation:

x2 - 6x + 8 = 0

Problem # 3:

Using the quadratic formula, again solve for x2 - 6x + 8 = 0

#### Part II

Now let's learn to graph quadratic equations. Do you want to learn a math rule that will make your life so much easier? Of course you do, and the rule is the graph of a quadratic equation is always some sort of parabola.

Problem # 4:

Graph the quadratic equation:

y = x2 - 1

### Step Two

Let's study a polynomial function.

Problem # 5:

Find both the degree and the leading coefficient of the polynomial:

3x2 + 2x3 - x + 5

### Step Three

Let's learn to graph both piecewise functions and rational functions.

#### Part A

Problem # 6:

Let's graph the piecewise function:

y= x + 2, x < 5

y = x - 4, x is greater than or equal to 5

#### Part B

Problem # 7:

Let's graph the rational function:

3x / (x + 1)

### Step Four

What are some of the properties of exponential functions and logarithmic functions?

Well, an exponential function is one in which the exponent consists of a variable.

y = 3x

y = 6x

Probably the two most common examples of rapid exponential growth are bacteria and bunnies reproducing rapidy.

On the other hand, a logarithmic function is basically the inverse of an exponential function.

y = log3 (x)

y = log6 (x)

One common example of logarithmic growth is that when someone starts to lift weights they improve rapidly, but later the growth is slower. A second example is how fast a beginner student learns to read, but this process slows in later years.

And of course, the graphs of exponential functions and logarithmic functions tend to look the opposite of one another. The former appear to start out slowly and then rise rapidly, while the latter tend to increase rapidly at first and then level off later.

Problem # 8:

Solve:

y = 3x * 6x

If:

x = 2

## Deliverable

Now it's time to demonstrate your command of this new precalculus material by choosing two of the following five presentations:

• Create questions and/or puzzles for your own mathematics quiz show, following the format utilized by Wheel of Fortune, Jeopardy, or another popular game show.
• Draw or construct relevant graphs in creative ways by using string, wool, or similar materials.
• Make a synoptic poster that contains all the information covered in this assignment, and use this poster to answer a series of questions where appropriate.
• Write a series of questions for fellow students to assess their understanding of these concepts and processes.
• You go ahead and ''be the teacher'' and walk through concepts and skills in this assignment in your own YouTube video or TED Talk podcast.

Solution # 1:

First, let's complete the square.

Solve for x2 - 6x + 11 = 0

Well, your standard form for a quadratic equation is:

ax2 + bx + c = 0

However, by completing the square we will change the equation to the form:

a(x + h)2 + k = 0

Furthermore, we know 11 is a prime number with its two factors 1 and 11, and we can't add 1 to 11 to get 6, so we can't factor this equation.

First, subtract 11 from both sides:

x2 - 6x + 11 - 11 = 0 - 11

To get:

x2 - 6x = - 11

Okay, now we need to compute the number which will complete our square.

Divide your b, which is -6, by 2, and you get -3.

Square your -3 to get 9.

You found it! Now add your 9 to both sides.

x2 - 6x + 9 = -11 + 9

Hey! Now you can factor the left side:

(x - 3)2 = -2

And finally:

(x - 3)2 + 2 = 0

Solution # 2:

Using factoring, solve the quadratic equation:

x2 - 6x + 8 = 0

Let's see, what numbers add up to -6 and multiply to equal 8?

(x +/- ?) (x +/- ?) = 0

We could choose:

1 and 8

Or:

2 and 4

It looks like we need to choose the latter:

(x - 2) (x - 4) = 0

Therefore, x = 2 or x = 4

Check your answers to see if you got the signs correct:

(x - 2) (x - 4) = 0

x2 - 4x - 2x + 8 = 0

x2 - 6x + 8 = 0

Ta-da!

Or:

Ta-dah!

Solution # 3:

Using the quadratic formula, again solve for x2 - 6x + 8 = 0

Do you remember your old reliable friend the quadratic formula?

If ax2 + bx + c = 0, then:

x = (-b +/- √(b2 - 4ac)) / 2a

First, rewrite your equation as:

(+1)x2 (- 6)x (+ 8) = 0

Now plug in your a, b, and c values:

(- -6 +/- √(-62 -4(1)(8)) / 2(1)

(6 +/- √(36 - 32)) / 2

(6 +/- √4) / 2

(6 +/- 2) / 2

(6 + 2) / 2 or (6 - 2) / 2

8 / 2 or 4 / 2

4 or 2

Therefore, x = 2 or x = 4

(+1)x2 (- 6)x (+ 8) = 0

Plug in 2:

(+1)(2)2 (-6)(2) (+8) = 0

4 - 12 + 8 = 0

12 - 12 = 0

0 = 0

That one works!

Plug in 4:

(+1)(4)2 (-6)(4) (+8) = 0

16 - 24 + 8 = 0

24 - 24 = 0

0 = 0

That one works as well!

We are so good at this precalculus thing today!

Solution # 4:

Graph the quadratic equation:

y = x2 - 1

First, plot some points:

If x = -2, then y = 3

If x = -1, then y = 0 (an x-intercept)

If x = 0, then y = -1 (the y-intercept)

If x = 1, then y = 0 (the other x-intercept)

If x = 2, then y = 3

Do you have them all plotted? Great! Now just smoothly connect them to draw your parabola. Do you see how (0, -1) is the bottom or tip or vertex of your parabola, and how it opens facing in an upward direction? Well done!

Solution # 5:

Find both the degree and the leading coefficient of the polynomial:

3x2 + 2x3 - x + 5

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