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Math 12 Advanced
Quadratics Test Review
It all started with sequences….
Arithmetic (linear function): d1 is constant, general term is t n  t1  (n  1)( d )
Quadratic (power sequence): d 2 is constant, d 2  2a (n increases by 1), general term is t n  an 2  bn  c
Regression: R 2  1 indicates a perfect fit (need to write out the screen that you see on calculator…..)
Forms of the Quadratic Function….
General Form: y  ax 2  bx  c, a  0

“a” is the vertical stretch, ”c” is the y intercept, line of symmetry x 
b
2a
Standard Form: y  a( x  h) 2  k
 “a” is the vertical stretch, vertex ( h, k ) , line of symmetry x  h
Transformational Form:
1
a
( y  k )  ( x  h) 2
Root and Stretch Form: y  a( x  r )( x  s )
 “a” is vertical stretch (a  0) , and r and s are the zeroes of the function
How to determine a vertex:

complete the square, x 
b
then substitute into equation to solve for y, calculator
2a
When to calculate a vertex;
 to maximize or minimize a value (maximum area, maximum revenue…)
How to determine the zeroes, roots or x intercepts:
 set equal to zero and factor or quadratic formula or complete the square or calculator
When to determine the zeroes, roots or x intercepts:
 solving a quadratic equation for x intercepts (y = 0)
 to solve for x when given a value for y
 distance, speed and time, revenue, “picture frame” questions,
Can You…
 Differentiate between vertex and solving questions?
 Determine line of symmetry if you know the x intercepts?
 Rearrange a rational equation in order to solve it?
 Determine an equation if you know the vertex and a point, the x intercepts, three points?
 Describe the nature of the roots?
 Isolate the a, b and c in a messy quadratic equation? ( 3x 2  2k  4 x  kx2  5  0 )
 Use the graphing calculator for regression, max and min, zeroes etc…?
 Complete the square and change between forms of the quadratic equation?
 Properly write out a complex number?
 Factor, use the quadratic formula and the graphing calculator?
Time to Practice…these questions are in no specific order – do you know the process needed?
 also questions on my web page under the exam prep stuff
1. Determine the vertex of the following functions – try to use two different methods.
a) y  2 x 2  3x  1
b) f ( x)   32 x 2  12 x  1
2. A baseball is thrown from a building and follows a parabolic path described by h  5t 2  5t  30 ,
where h is the height in metres and t is the time in seconds.
a)
b)
c)
d)
e)
f)
g)
What is the initial height of the ball? How do you know?
When will the ball hit the ground?
When does the ball reach the maximum height?
What is the maximum height?
When is the ball at a height of 31metres?
How high is the ball 1.6 seconds after it is thrown?
Domain and Range
3. Bill and his sister have to cut the lawn. The lawn is a square and Bill says he will cut an even strip
around the outside edges that is 3m wide and his sister will cut the rest. If they both cut the same area of
lawn, what were the original dimensions of the lawn?
4. Two numbers differ by 11. Find the values of the two numbers if the sum of their squares in a
minimum.
5. Find two numbers whose sum is twenty and whose product is a maximum.
6. Find the quadratic equation for a parabola with a vertex of (-1, -4) and passing through (3, 4).
7. Find the quadratic equation for a parabola with x-intercepts of (-1, 0) and (5, 0) that passes through
point (3, 16).
8. Find the value of k if the following function has no real roots: y  kx2  4 x  3
9. A senior’s dance club has a $3 cover charge and averages 300 customers on a Friday night. They
discovered that for every increase of $0.50 in cover charge, the number of customers decreases by 10.
Find the cover charge that maximizes revenue.
10. The underside of a bridge forms a parabolic arch that is 32m wide at the water level and 12m high at
the center.
a) Find the quadratic equation representing the underside of the bridge.
b) How far apart are supports that are 8m tall?
11. Mary drove from Dryden to Winnipeg, a distance of 500 km. She then increased her speed by 20 k/hr
for the 720 km trip from Winnipeg to Regina. If the total trip took 11 hours, what was her speed from
Dryden to Winnipeg?
12. A quadratic function has a leading coefficient of -2, a constant term of 6 and no x term. Find the zeros
of the function.
13. Determine a quadratic equation or two for the following roots.
a) 3 and -2
d)  2
c)  5i
b) ½ and ¾
e) 1  i 3 (tricky)
14. Describe the nature of the roots…remember to set equal to zero!
b) x 2  8 x  16
a) y  3x 2  4 x  2
c)
2
5

3
x 1 x  6
15. Solve the following – try a couple of different methods. (leave answers exact)
a) 42  x 2  x
b)
2  x 3(4  x)

x
2 x
c)  2( x  2) 2  10  y
d)
4x
3

1
x 1 x  2
16. Twice a number is added to 3 times its square and the result is 385. What is the number?
17. The flight of a football is given by the equation, h(t )  24t  5t 2 , with h in meters and t in seconds.
a) Algebraically calculate the time required for the ball to reach a height of 10m.
b) How long will the ball be above a height of 10 m?
18. Solve for x: ( x  m) 2  ( x  m)  12 (hint: let k = x – m)
19. Consider the following sequences. If arithmetic or quadratic, state why and generate the general term.
a) {4, 7, 10, 13, 16…}
b) {2, 4, 8, 16, 32…}
c) {4, 13, 26, 43, 64…}
20. Given, x 2  kx  5  3k , determine value of k such that…
a) 2 equal real roots
b) 2 real distinct roots
c) no real roots
Solutions:
1. a)
 34 , 81 
b) (4, 23)
2. a) 30m b) 3s c) 0.5s
d) 31.25m
4. 211 and
3. 14.49m by 14.49m
e) 0.28s and 0.72s
11
2
f) 25.2m
5. 10 and 10
g) D: [0, 3] R: [0,31.25]
6. y  12 ( x  1) 2  4
7. y  2 x 2  8x  10 (use root and stretch formula – put in roots and point as x and y and solve for a)
8. k  4/3
9. $9.00
13. a) y  x 2  x  6
14. a) 2 non real
10. a) y 
3
64
( x  16) 2  12 b) 18.48m
b) y  8 x 2  10 x  3 c) y  x 2  25
b) 2 real and equal
17. a) 0.46s b) 3.88s
d) y  x 2  2
c) 2 real and distinct
15. a) x = 7, x = -6 b) x  3  7 c) x  3  5 d) x 
11. 100km/h
12.  3
e) y  x 2  2 x  4
(be able to state why for each of these)
2 i 11

3
3
16. 11 or -35/3
18. x = m + 4, x = m – 3 19. a) d1 constant, arithmetic, t n  3n  1 b) neither
c) d 2 constant, quadratic, t n  2n 2  3n  1 20. a) 10 or 2 b) k: (-, 2)(10, ) c) k: (2, 10)