SUMMER WORK FOR STUDENTS TAKING PRECALCULUS IN 2016-­‐17
Barrie Prep
Summer 2016
● Solve part I in June, part II in July, part III in August. The idea of the review is to do little work (each of these 3 packets should take about 3-­‐4 hours of work), but spread in time so it refreshes your memory every now and then and keeps alive the knowledge you acquired during the year. Of course, you can do everything in August, but by then your memory will be stale beyond the rejuvenating power of this packet. Or you can do it all now, but by late August, when classes begin, you will have trouble to retrieve this knowledge from the dungeons of your memory. If you want to invest your time poorly, go ahead.
● If you solved a problem and the solution can be checked using your calculator, by all means do it. Moreover, consider the checking-­‐with-­‐your-­‐calculator thing part of the problem.
● You have hints for most of the groups of problems. Use them wisely: try to work out the problem on your own first. If you see that you cannot, go to the hints. And, if you solved the problem but your calculator seems to have a different opinion about the solution, in that case too, go for the hints. Or if your solution should be Baltimore and you got Anchorage, in that case too, use the hints.
● Work hard at resolving discrepancies. Not so hard that it mars your vacation (intelligence includes having a sense of proportion).
● If a problem or group thereof is still driving you crazy, you will find help at [email protected]. USE IT! Be explicit about what you did, what you got, and/or what you find puzzling. A couple of sentences. You may include a scan in your email. Help may delay a day or two.
● Turn in the packet with your solutions the first day of classes. It will be a conversation piece the first week of class, and part of your first quarter grade.
● Enjoy your vacation!
Part I – June
1. Graph each of the following functions without the aid of the graphing features of your calculator. USE THE MOST EFFICIENT METHOD 1a. Remember to find and label x and y intercepts and vertex.
1b. Remember to find and label x and y intercepts and vertex.
1c. Remember to find and label x and y intercepts and vertex.
2. Complete correctly. Yes, simplify as much as you can too
2a. 2b. 2c. 3. 3a. 3b. 3c. Solve the following using the most efficient method
4*. Solve the following inequalities. Graph!! Include any graphs you use and your reasoning. Avoid the use of the graphing features of your calculator, if possible.
4a. 4b. 5. 5a. 5b. 5c. 5d. Reduce to one single exponential:
6. 6a*. 6b. Find the inverse of each of the following functions
6c.* 7a. The graph of a function f is displayed below. Draw the graph of different color in the same system of axes.
with a 7b. The graph of a function f is displayed below. Draw the graph of different color in the same system of axes.
with a 8. The current population of India is estimated to be 1.27 billion, and growing at the annual rate of 1.22% (CIA World Factbook)
8a. Write a function P(t) that expresses the approximate size of the population of India t years from now, assuming the trend does not change.
8b. What will the population of India be in 2019?
8c. When would you expect the population of India to reach 1.4 billion?
P(t) = ………………………………..
Part II – July
1. Match each pair of rules to one pair of graphs. Justify your choices
A
B
D
E
E
a. b. c. d. e. f. C
F
2. 2a. 2b. 2c. Solve the following equations. Check your solutions to make sure they work.
3. Find plausible rules for the following graphs:
3a. f(x) = …………………………………….
3b. 3c. f(x) = …………………………………….
f(x) = …………………………………….
4. Graph the following functions without the aid of the graphing features of your calculator. Remember to find intercepts, important points if any, long term behavior if necessary (touchdown, ballet, asymptotes), and explain your reasoning.
4a. 4b. 4c. 5. Write two vertical shifts of an exponential rule: 5a. One for a graph that is increasing and does not cross the x-­‐axis
5b. Another one for a decreasing graph that crosses the x-­‐axis.
F(x) = …………………………………
G(x) = …………………………………
6. In 1987, the life expectancy of a baby born in a low income country was 54.8 years. Improvement of life conditions and advances in medicine are the main factors that contribute to an increase of the life expectancy of about 1.2 years for every 5 years that elapse.
6a. Complete the following table.
Number of years after 1987
Life expectancy
0
5
10
12
15
18
20
6b. Find a function that expresses the life expectancy E(t) of a baby born in a low income country t years after 1987, assuming that the trend does not change.
E(t) = ………………………………………………….
6c. What life expectancy will you estimate for a baby born in a low income country in the year 2020?
6d. According to your model, when will the life expectancy in low income countries reach 65?
7. Graph the function Make a sign chart
Mark and label x-­‐intercepts and shade regions
Study tail (long term behavior) Graph
8. 8a. 8b. Solve the following inequalities (use your calculator only as last resort):
9. Graph the functions and in the same system of axes. Use different colors. Find intercepts and other important points for each function individually and then, find the points where the two graphs meet. Label all those points on your graph.
Part III -­‐ August
1. Graph the following functions without the aid of the graphing features of your calculator. Remember to find intercepts, important points (e.g. vertex) if any, long term behavior if necessary (touchdown, ballet, asymptotes) and explain your reasoning.
1a. 1b. 1c. 2. The graph of the function f appears below. 1
In each of the following cases, use the system of axes given to graph the indicated function:
2a. 1
-3
3
2b. 1
-3
3
2c. 1
-3
3
2d. 1
-3
3
2e. 1
-3
3
3. 3a. 3b Solve the following equations
3c. 3d. 4. The following graph (with some more detail) appears on aerobic equipment in practically all the gyms to describe the average anaerobic threshold (number of heart beats per minute for trained muscles to burn glucose with oxygen ‘debt’) 4a. Use point-­‐slope to find the rule that connects the anaerobic threshold and age:
4b. What is the anaerobic threshold for a trained 50 year old?
y = f(x) = ……………………………..
4c. During a training session, a football player has a heart beat of 164 and the team doctor tells him that he has not reached his anaerobic threshold yet. What can you say about his age?
5. 1 3 Match the pair of graphs to the pair of rules. Explain your reasoning.
5 2 4 6 6. Graph the function
7. 7a. 7b. 7c. 7d. Solve the following inequalities
8 Graph the following functions
8a 8b.* 8c.* 9. 9a. 9b. 9c. 9d. Find the following values of ln without the aid of technology. Show your work.
Part I -­‐ Hints
Problem 1 Hint
Quadratic functions come in three forms: ● Standard form (use quadratic formula to get x-­‐int, remember that quad form also gives you the x of the vertex)
● Factored form (use the zero product property to get x-­‐int, use symmetry to get x of vertex)
● Shift form (discover the shifts to place the vertex – undo operations to get x-­‐int)
● Note: the zero product property is that for a product to be 0 either the first factor or the second factor must be 0, or both.
Problem 2 Hint
● For 2b: When you have square minus square, you can rewrite that as a sum (of the numbers under the squares) times a difference (of the same numbers): ● For 2c: Remember that Problem 3 Hint ● Quadratic formula works only when one of the sides is 0
● The property of the zero product works only when we have product = zero
Problem 4 Hint ● When you have to solve an inequality, graph!
● 4b. One possibility is to leave 0 on one side (e.g. subtract 2x from both sides) and proceed as before
● You can always graph 2x ≥ 3x2 – 1 one in red, the other one in blue, ask yourself which one you want to be bigger, then find the good x’s. ● Last resort: You may use your calculator.
Problem 5 Hint ● exponentials with the same base, multiplied, you get another exponential with the same base, exponents added
● exponentials with the same base, divided, you get another exponential with the same base, exponents……………… (complete it yourself!)
● negative exponent is like 1 over, 1over is like a negative exponent.
● exponential of an exponential you multiply the exponents
● ● BTW, so Problem 6 Hint ● Replace f(x) with y, then undo operations to find x.
● Logarithms undo what exponentials do. Exponentials undo what logarithms do
● When the moment arrives to undo a log or ln, you exponentiate both sides, e.g.
(for ln) or (for log)
● To undo reciprocation you reciprocate both sides.
● LHS = left hand side, RHS = yeah
Problem 7 Hint ● The absolute value positivizes the values of the function.
Problem 8 Hint ● 8a: Go one year forward, first. Do not forget the idea of ‘old people’ and ‘new babies’.
● There are two quantities connected, P and t. Is 1.4 billion a value of P or of t?
● 8c: Set up an equation first. If you use graphing, graph Y = the population function and Y = 1.4 together.
● Run some calculations to make sure that the result is reasonable (if, for 8c, you get tomorrow or in the year 3000, you know that it is wrong).
Part II -­‐ Hints
Problem 1 Hint Choose a graph ● What do you know about the slopes?
● Which pairs of rules have suitable slopes?
● What do you know about the y-­‐intercepts?
● Which pairs of rules have suitable y-­‐intercepts?
If there is still more than one possibility, combine slope and y-­‐intercept e.g.
the less steep line has
bigger y-intercept
the steeper line has
smaller y-intercept
Problem 2 Hint
● To ‘undo’ the absolute value think: ‘For of stuff’?’ if any.
, what are the possible valueS either stuff = 10 or stuff = –10
● Sometimes there are no solutions because an AV cannot be negative.
Problem 3 Hint ● For straight line: Slope is rise/run.
● For the quadratics: What do you know about the function?
● If you have the x-­‐intercepts, use factored form (don’t forget the leading coefficient!)
● If you have the vertex, use shifts. (Again, don’t forget the leading coefficient!)
● y-­‐intercepts: write them as x y and use them to find a
0
Problem 4 Hint ● Exponential graphs look like planes taking off (if base > 1 and exponent is x or base < 1, and if exponent is –x = negative x)…
● …or landing on the x-­‐ axis (if base < 1 and exponent is x or base > 1, and if exponent is –x = negative x)
● Vertical shifts move the ‘runway’ of the plane up or down. Draw the runway (i.e. the asymptote) before the graph.
● When there is a vertical shift you may have to find an x – intercept.
Problem 6 Hint ● How much would you say life expectancy grows each year (rise for a run of 1)
● What type of function (linear, quadratic, exponential, log) do you have?
● For parts c and d, the value I have, is it a value of t or a value of E?
Problem 7 Hint
o Make a sign chart sgn (2 – x) draw y=(2 – x) dotted
sgn
draw y=(x2 – 5x – 6)
sgn f(x) ● Study tail (long term behavior). If you multiply out, do you get basically
, what?
● The long term behavior can be like or . (carrying suitcases!!)
Problem 8 Hint
● Inequalities: graph! graph!
● What are the good values of x?
(touchdown ). Like
. (Ballet?). Or like ● Find the numbers that separate the good values of x from the bad ones. In 8a, use last-­‐in-­‐first-­‐out for that.
Part III – Hints
Problem 1 Hint
I hope you remember what linear, quadratic and exponential graphs look like. For help with exponentials, go to Hints for Part II, problem 4
Problem 2 Hint All these are shifts, reflections, absolute values, or multiplications by a number of a function f. Think of the effect of each on the graph, move a few well chosen points of the graph according to that effect, then move the entire graph.
Problem 3 Hint
● 3a, 3b Think quadratic
● 3c. Use the zero product property AND then find x (what undoes what an exponential does?)
● 3d. Undo operations (what undoes log? Answer: exponentiating both sides with base 10)
Problem 5 Hint
● What do you know about the vertices? Notice that you may have to work to find the vertices in some cases
● What do you know about concavity? (smiling/frowning)
● What do you know about intercepts?
● Combine these things e.g. ‘the smiling one has a positive x intercept and a negative one, the y intercept is positive, the vertex is blah, blah, blah’
Problem 6 Hint
● Make a sign chart.
● You cannot deal with third degree polynomials. Factor x out
● You will have three factors: x, (
), and (
), so your chart must have four lines: one for each factor and one for the function.f
● Remember to get long term behavior
Problem 7 Hint
● Do I have to say it again. GRAPH! Choose two pretty color markers and graph the LHS and the RHS. Then find the goooood values of x. Problem 8 Hint
● What does the graph of ln x look like? If in doubt, use a transparency: ln is the inverse of the exponential with base e
● 8a. You have replaced x with (x – 1) in ln x. What does that do to the graph?
● 8b and 8c Use shifts, reflections, scaling of 1/x and 1/x2 resp.
Problem 9 Hint as , ln undoes what the exponential base e does, so if you rewrite