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.
Answers
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
Check your answers:
(+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
Did you answer 2 as your degree and 3 as your leading coefficient? Well, this was a trick question, and that is incorrect.
Why? A polynomial in standard form must be written with the highest degree to the left, and then with the exponents in decreasing order, with the constant to the far right.
So we rewrite our equation in standard form as:
2x3 + 3x2 - x + 5
And now we see our correct answer is a degree of 3 and a leading coefficient of 2 instead. In the future, always make sure your polynomial is in the correct order as dictated by the standard form.
Solution # 6:
Let's graph the piecewise function:
y = x + 2, x < 5
y = x - 4, x is greater than or equal to 5
''Hey! Are you trying to trick me again? This function has two different parts.''
Yes, I'm trying to trick you again. That's what I love to do every day, trick math students with tricky problems. Just kidding! This isn't too bad, if you graph the two parts separately.
Let's graph the first part, which is:
y = x + 2, x < 5
Pick your points:
If x = -2, then y = 0 (the x-intercept)
If x = -1, then y = 1
If x = 0, then y = 2 (the y-intercept)
If x = 1, then y = 3
If x = 2, then y = 4
If x = 3, then y = 5
If x = 4, then y = 6
If x = 5, then y = 7
Hey! Wait a minute! Our rule from above states that x < 5, so we can't graph this last point. Put an open circle at (5, 7) instead.
Go ahead and connect your points with your ruler, as you can see this graph is a straight line segment.
Next, let's graph the second part, starting at 5, which is:
y = x - 4, x is greater than or equal to 5
If x = 5, then y = 1
If x = 6, then y = 2
If x = 7, then y = 3
If x = 8, then y = 4
If x = 9, then y = 5
Again, this is a straight line segment, so connect your points. However, the point (5, 1) should be a dark circle instead of an open circle.
You have now completed the graph of your piecewise function, which consists of two different line segments.
Solution # 7:
Let's graph the rational function:
3x / (x + 1)
Pick your points:
If x = -3, then y = 4.5
If x = -2, then y = 6
If x = -1, then y = undefined (you can't divide by zero)
If x = 0, then y = 0
If x = 1, then y = 1.5
If x = 2, then y = 2
If x = 3, then y = 2.25
This graph is going to have an asymptote at. . . you guessed it, x = -1.
Go ahead and plot your points, and then smoothly connect them to create your two separate curves.
Lastly, Google this equation to see the correct graph by double-checking your answer.
Solution # 8:
Solve:
y = 3x * 6x
If:
x = 2
y = 32 * 62 =
9 * 36 =
324
Assignment Rubric
| Requirements |
0 - 5 points |
| Problem # 1 is solved correctly |
|
| Problem # 2 is solved correctly |
|
| Problem # 3 is solved correctly |
|
| Problem # 4 is solved correctly |
|
| Problem # 5 is solved correctly |
|
| Problem # 6 is solved correctly |
|
| Problem # 7 is solved correctly |
|
| Problem # 8 is solved correctly |
|
| Deliverable # 1 is clear and concise |
|
| Deliverable # 2 is clear and concise |
|
| Total: |
/ 50 points |
