Math 12 Extended Response Exam Review
Section One: Quadratics
1. Given the quadratic function :
(a) Algebraically determine the coordinates of the vertex of the parabola represented by the function above.
(b) State whether the vertex in (a) is a maximum point or a minimum point.
2. Given the graph to the right, do the following
Tasks without using the regression feature
On your graphing calculator.
(a) Determine the transformational form
Of the function represented by the above
Graph.
(b) One the same grid, trace a parabola
That has the same x-intercepts as the
given parabola and a maximum value of 20.
Write the coordinates of the vertex
and 3 other points on this parabola.
3. Solve algebraically to find the exact roots of the following equations. Simplify where possible.
(a) (b)
(c) (d)
4. A snowball is thrown into the air. The function expresses the relationship between height, h, in metres and time, t, in seconds.
(a) Algebraically determine the maximum height the snowball reaches.
(b) How long is the snowball in the air?
5. A rectangular rink with dimensions of 25 m
by 20 m is to be expanded by adding a
rectangular strip of uniform width as shown
to the side. If the new rink is to have an area
of 644 m2, what will be the width of the strip?
6. Bill kicks a football in Tom’s direction. The football follows a parabolic path. Tom,
who does not know it has been kicked, may be standing in the football’s path. After
having travelled a horizontal distance of 10 m, the football reaches a maximum height of
18 m. Will Tom, who is 1.8 m tall, get hit by the football if he’s standing 19.8 m from where
the football was kicked? Solve this problem algebraically.
7. (a) Write the function, in transformational
for or standard form, that represents the
following parabola.
(b) State the domain and range of the function.
8. Calculate the discriminant of and explain what the result tells you about the graph of .
9. The severity of an automobile crash increases significantly as the speed increases. The table shows the relationship between the speed and a crash severity index.
(a) Jimmy claims that a quadratic function would best model this situation. Is Jimmy’s claim correct? Explain.
(b) What speed would have a crash severity index of 50.40?
10. Solve for “x” in each of the following equation using a different algebraic method for each.
(a) (b) (c)
(d) (e)
11. A football is kicked into the air. The equation expresses the relationship between height, h, in metres and time, t, in seconds.
(a) Determine the maximum height reached by the football.
(b) At what time(s) after the kick is the ball at a height of 5m?
12. A golf ball is hit from ground level in a flat field and reaches a maximum height of 25 m. The ball first hits the ground 100 m away while following a parabolic path.
(a) Draw a diagram and include all your important information needed to model this problem. Remember to label your axis.
(b) How high is the golf ball above the ground with it is at a horizontal distance of 20 m from where it was hit?
13. Given the graph to the right, do the
Following tasks without using the regression
Feature on your graphing calculator.
(a) Determine the general form of the
Function represented by the graph to the
Right.
(b) One the same grid, trace a parabola that
Has the same x-intercepts as the given
Parabola and a maximum value of 8. Indicate
The coordinates of the vertex and 2 other
Points on the curve.
(c) Write an equation of another quadratic function that has the same zeroes as the function shown above.
14. The number of dots in each of the following figures forms a sequence.
(a) Determine the function that generates the sequence.
(b) If the figures continue to follow the same pattern, which figure would contain exactly 590 dots?
15. Analyze the function by identifying its i) Vertex ii) y-intercept iii) domain iv) range v) axis of symmetry vi) zeroes.
16. Given the function .
(a) Show how you would change the function into transformational form.
(b) Describe the steps you would take to sketch the graph using the transformational form of the equation.
(c) Graph the function
17. Given the following table of values:
X / 1 / 2 / 3 / 4 / 5 / 6Y / 5 / 19 / 43 / 77 / 121 / 175
(a) Is the relationship between the x and y values exponential or quadratic? Explain
(b) Find the equation that represents the relationship.
18. At the Halifax Airshow, a plane performs a power dive. The equation expresses the relationship between height, h, in metres and time, t, in seconds.
(a) What is the minimum height that the plane reaches?
(b) When will the plane be at a height of 35 m?
19. A rectangular field measures 30 m by
60 m. A strip of uniform width (x), as shown
in the diagram below, is mowed so that the
area of the uncut field is 70% of the original
area. What is the width of the strip that has
been mowed?
20. Given the function (a) Write the function in standard or transformational form. (b) What is the vertex of the parabola? (c) What is the equation of the axis of symmetry?
21. A ball was released and rolled down an inclined plane.
Its distance with respect to time since released is recorded in the following scatter plot. It was determined that a quadratic equation would best represent this data.
(a) Using your graphing calculator, find the equation of best fit and fill in the values for a, b, c, and . The quadratic equation is: ______
(b) What is the significance of the value obtained?
(c) Given the ordered pair (6, ?), determine the missing coordinate. What does this ordered pair represent in the context of the given problem?
22. Solve the following equation. If the root(s) are non-real, express in terms of . .
23. A photograph measures 40 mm by 62 mm. A frame of uniform width is placed around the photograph, doubling the area. What is the width of the frame?
24. A ball is hit and its height “h” in metres, with respect to time, “t”, in seconds, is expressed by
(a) What was the initial height of the ball when it was hit?
(b) What is the maximum height of the ball and what is the time required for the ball to reach its maximum height?
25. Given (a) Solve for “x” (b) Solve for “x” using a different method.
26. Write two different quadratic functions such that each has x-intercepts at -5 and 4.
27. The function describes the height of a baseball “h”, in metres, as a function of time “t”, in seconds, from the instant the ball is hit.
(a) Express this function in transformational form.
(b) How long will it take the baseball to reach its maximum height?
28. You want to put a fence around your garden. One side boarders your house, so it does not need a fence. If you only have 50 m of fencing, what dimensions should you make your garden in order to maximize the area?
29. A rectangular driveway measuring 8 m by 6 m was reduced in size such that the new area is 70% of the original area. To accomplish this, a strip of uniform width was removed from ONE end, and a strip of the same width was removed from ONE side. What was the width of the strip?
30.A lifeguard is roping in a “safe swimming area”
at the beach where they currently work. They want to divide the safe swimming area into a section for small children and a section for large children. The lifeguard has 120 m of rope. One side is the beach, therefore, does not need a rope. (See the diagram to the right).
(a) Express the entire area of the swimming sections as a quadratic function in terms of either “x” or “y”.
(b) Use the quadratic function obtained in (a) to determine the length and width that will produce a maximum swimming area.
Section Two: Exponents and Logs
1. Algebraically solve for x:
(a) (b) (c)
(d) (e) (f)
(g) (h)
(i) (j) (k)
(l) (m) (n)
(o) (p) (q)
(r) (s)
2. Michael was running a Biology experiment where he was determining and recording the approximate number of bacteria over time. The following is a partial record of some of the readings.
No. of Hours / 0 / 2 / 4 / 6 / 8 / 10 / 12No. of bacteria / 100 / 800 / 1600
(a) What is the initial number of bacteria in the culture?
(b) Determine an equation to calculate the number of bacteria at any time.
(c) What is the bacteria count after 20 hours?
3. Suppose the cost of a parking permit increases by 5% yearly. If the cost of parking is now $300 per year, how long will it take for the price to increase to $400 per year?
4. (a) Describe in words how the graphs of and for b>0, and b 1 are related. You must state a total of 3 similarities and/ or differences.
(b) Given the function , for what values of “a” and “b” will the graph of the function be an exponential growth curve?
5. Susan tried to solve the equation . She got the error message “NONREAL ANS” on her TI-83 calculator when trying to evaluate . Explain why.
6. When Drug 1 enters the bloodstream, it gradually dilutes, decreasing exponentially, by 20% every 5 days. A second drug, after entering the bloodstream, also decreases exponentially, but only by 10% every 7 days. If the initial amount of Drug 1 is 200 mg and the initial amount of Drug 2 is 150 mg, create and use functions to determine which drug has the greater amount remaining after 12 days.
7. Given
(a) Determine the coordinates of the y-intercept.
(b) Write the equation of the horizontal asymptote.
(c) Indicate whether the function above represents a growth curve or a decay curve. Explain how you know.
8. Express the following expression as a single logarithm. Simplify your answer. .
9. Evaluate:
(a) (i) (ii)
(b) (i) (ii)
(c) Based on the answers obtained in parts (a) and(b), write an expression equivalent to
10. Marla tries to evaluate (Zero to the exponent negative 3) on her calculator and gets an error message. Explain why she gets an error message.
11. Describe a situation that could be modelled by the function
12. Show how to evaluate the expression , without the use of a calculator.
13. A general rule used by car dealerships is that the trade-in value of a car decreases by 30% each year.
TTime in years / V
Value of the car
1
2
3
(a) Suppose that you own a car whose trade-in value, V, is presently $3570. Determine how much it will be worth one year from now, two years from now, three years from now. Fill in the table of values.
(b) Without using the equation of the function, explain why an exponential function can be used to represent the data above.
(c) Write the particular equation expressing the trade-in value, V, of your car as a function of the number of years, t, from the present.
X / 0 / 2 / 4 / 6Y / 4.7 / 14.1 / 42.3 / 126.9
14. Find the equation of the function represented
in the following table. Do not use a graphing calculator.
15. At the start of the year 2004, Jonathon invested $500 in a fund that doubles every seven years. In what year will he have $1200 in his account?
16. Evaluate the following without using a calculator. Show at least one intermediate step needed to obtain your final answer.
(a) (b) (c) (d)
17. Is a geometric sequence? Explain your reasoning.
18. (a) Without the use of the regression feature of the graphing calculator, determine the exponential equation that is represented by the data in the following table.
X / -3 / 0 / 3 / 6 / … / 8Y / 15 / 12 / 9.6 / 7.68 / …
(b) Find the missing value in the table to the right.
19. A certain bacterial culture initially has 200 bacteria/ cm2 and the number doubles every 20 minutes. (a) Find the equation representing this situation. (b) How many bacteria/ cm2 would there be after 4 hours? (c) How long would it take until there are 1000 bacteria/ cm2?
20. If , solve for x.
21. The population of a newly discovered organism can be described by the function where “P” is the number of organisms and “t” is time (in minutes).
(a) In the equation , what do the “3” and the “10” signify in the context of the problem?
(b) How long does it take the population to double?
22. Show that is a geometric sequence for all values of “a” and “d”.
23. Using the laws of logarithms and the definition of an arithmetic sequence, show that is an arithmetic sequence.
Section Three: Circle Geometry