All graphs must be drawn to scale with a ruler (no rough sketches). Key points should be plotted accurately and joined with a smooth curve. Clearly and neatly label the x and y axis. All answers to questions must be written in complete sentences.
Each question will be evaluated for all three reportable standards, concepts and procedures, problem solving, communication and reasoning.
1. Functions
A. In what four ways can a mathematical function be represented?
B. Give/create/find an example of a function and represent it in all four ways.
C. Which one of the four ways do you prefer and why?
2. Exponential Functions
A. As a promotion, a clothing store draws the name of one of its customers each week. The prize is a coupon for 20aed for use in the store. If the winner is not present at the drawing, he or she cannot claim the prize and the amount of the coupon increases for the following week’s drawing. The function f(x) = 20(1.2)x gives the amount of the coupon in dirhams after x weeks of the prize going unclaimed.
a. Create a table to record the weeks from 0 to 25 and the value of the coupon.
b. Draw a graph of the function when 0 ≤x≤25. The graph must be drawn to scale with a ruler. Key points should be plotted accurately and joined with a smooth curve. Clearly and neatly label the x and y axis.
c. Write a short paragraph stating the following: (Write in complete sentences)
i. initial value of the coupon
ii. the percent increase each week
iii. the amount of the coupon after 2 weeks of the prize going unclaimed
iv. the number of weeks after the prize going unclaimed when the coupon will be greater than 300 dirhams
v. how much it will be worth if it is not claimed in one year.
B. The function f(x) = 40(.8)x gives the amount of a medication in milligrams present in a patient’s system x hours after taking a 40mg dose. A patient can go back to work if he or she has less than 1 mg left in his system. The patient takes the medication at 8pm on Monday night. Explain when he can go back to work. Justify your answer by showing all the work.
3. Logarithmic Functions
A. Copy the tables on your own paper and complete:
x
y = log2x
-2
0
1/4
1/2
2
4
8
x
y = log x
-1
0
1/100
1/10
10
100
1000
B. Graph y = log2 x and y = log x on the same graph. Draw an x-axis from -1 to 10 and a y-axis from -10 to 10. Use a different color for each function. Label at least 2 points on each graph, including the x-intercepts. The graph must be drawn to scale with a ruler. Key points should be plotted accurately and joined with a smooth curve. Clearly and neatly label the x and y axis.
C. Identify the domain and range of the functions. Explain why you can’t take the logarithm of a negative number.
4. Conic Sections
A. Identify one real-world application for each:
circle
ellipse
hyperbola
parabola
You must identify the conic section and describe how the conic section applies to the application. This must be at least one page using 12pt font. You may include small pictures that have been cut and pasted or drawn, but the writing must be one page.
B. Write the equation of a circle, ellipse, and parabola with the center at (3, -5). Write an equation of a circle, ellipse, and parabola with the center at the origin. You may choose any of the other parts (such as the radius) necessary.
5. Systems of Equations
You are an entrepreneur and are just starting a company that will sell hoodies with the school logo. You must pay 4000aed for start-up costs and 35aed for each one you produce. You will sell them for 150aed each.
a. Write two equations: one for the cost and one for the revenue
b. Draw a graph with “Hoodies produced and sold” for the x-axis and “AED” for the y-axis. The graph must be drawn to scale with a ruler. Key points should be plotted accurately and joined with straight lines. Clearly and neatly label the x and y axis.
c. Label the point where you will break even on the graph.
d. Shade the area that represents the loss red and the area that represents the gain green
e. Answer the following questions in paragraph form:
How many hoodies do you need to sell to break even?
How much will you lose if you only sell 30 hoodies?
How much will you earn if you sell 50 hoodies?
6. Trigonometry
A. Solve each problem by drawing a diagram, writing an equation and solving the equation.
A wheelchair ramp is 5 meters long and rises 0.8 meters. What is the angle of elevation to the nearest tenth of a degree?
A 9-foot ladder rests against the side of a wall. The bottom of the ladder is 2 feet from the base of the wall. Determine the measure of the angle between the ladder and the ground. Round to the nearest tenth.
An airplane is flying 2 miles over the ocean directly toward a coastline. If the angle of depression to the coastline from the airplane is 15°, how much farther does the airplane have to fly before it is directly above the coastline? Round to the nearest tenth.
A piece of wire is attached from the top of a 45-foot utility pole to a stake in the ground. If the angle of elevation from the stake to the top of the pole is 39°, find the length of the wire. Round to the nearest tenth.
Create your own problem involving a right triangle and an angle of depression.
B. Right triangle ABC has a right angle at C. Explain how the cosine of A and the sine of B are related. You may want to use a diagram to help with the explanation.
