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MPM2D Mr. Jensen Name:______________________________
Chapter 6 Practice Test
Section 1: Multiply Choice [6 marks] 1) Determine the value of ‘c’ that makes x2+16x+c a perfect square trinomial.
a) 64 c) 8 b) 16 d) 32
2) Which graph represents a quadratic relation whose corresponding quadratic equation has no solutions?
a) b) .
c) d)
Standard Form 2y ax bx c= + +
Factored Form ( )( )y a x r x s= − −
Vertex Form 2( )y a x h k= − +
Quadratic Formula 2 42
b b acxa
− ± −=
3) State the x-‐intercepts of the quadratic relation .
a) (1,0) and (–2,0) c) (–2,0) and (0,0)
b) (–1,0) and (2,0) d) (–1,0) and (0,0)
4) Which statement best explains why there are 2 roots to the equation 𝑦 = 𝑥! + 12𝑥 − 36 ?
a) The value of 12! − 4(1)(−36) is negative.
b) 12! − 4(1)(−36) is a funny looking algebraic expression. c) The value of 12! − 4(1)(−36) is positive.
d) The value of 12! − 4(1)(−36) is zero.
5) What is the best description of the term ‘zeros’ for a quadratic relation?
a) the maximum or minimum points of a parabola
b) the shape of the graph
c) the vertical line containing the vertex
d) the x-‐intercepts of the graph
6) How many real roots can a quadratic equation of the form have?
a) no real roots c) two real roots
b) one real root d) All of these
Section 2: Completing the Square [4 marks]
2) Put each of the following parabolas into vertex form by completing the square. State the vertex and whether it is the maximum or minimum point of the parabola.
a) y = -‐ x2 + 6x − 1 ` The vertex is: ____________________ Is this a max or min point? ____________
b) y = 2x2 + 12x + 4
The vertex is: ____________________ Is this a max or min point? ____________
Section 3: Solve by Factoring [3 marks]
3) Solve each of the following by factoring:
a) x2 + 2x − 15 = 0
b) 4x2 -‐ 8x − 5 = 0
c) 3x2 − 11x = -‐8
Section 4: Solve Using the Quadratic Formula [3 marks]
4) Solve each of the following using the quadratic formula. Round your answers to the nearest hundredth. (make sure to clearly identify your solutions/x-‐intercepts):
a) -‐3x2 − 11x + 8 = 0
b) 4x2 − 6x + 9 = 0
c) 𝑥! + 7𝑥 = −5
Section 5: Write the Equation in Standard Form [2 marks] 5) For the following parabola: a) Write an equation in factored form, y=a(x-‐r)(x-‐s) :
Equation of the Parabola in Factored Form: _____________________________________ b) Write an equation in standard form, y=ax2+bx+c : (Hint: Use FOIL to expand the equation that is in factored form, this will give you standard form!)
Equation of the Parabola in Standard Form: _______________________________________________________
Section 6: Graph a Quadratic that is Given in Standard Form [4 marks]
6) For the quadratic equation 𝑦 = 2𝑥! + 𝑥 − 15: a) Find the x-‐intercepts (solve it using factoring or the quadratic formula!!) :
b) Find the axis of symmetry. (x-‐coordinate of vertex):
c) Find the vertex:
d) Sketch the graph of the parabola. Label the x-‐intercepts and the vertex.
Section 7: Application [2 marks]
7) The path of a soccer ball after it is kicked from a height of 0.5 m above the ground is given by the relation
, where h is the height above the ground and d is the horizontal distance, both in metres.
a) How far has the soccer ball travelled horizontally, to the nearest tenth of a metre, when it lands on the ground?
b) What is the maximum height reached by the soccer ball? At what horizontal distance is the soccer ball at its maximum height?
Bonus (+2)
1) Show the proof of the quadratic formula:
2) The cost, C, in thousands of dollars, to produce items at a clothing manufacturer is given by the relation 𝐶 = 2𝑥! − 29𝑥 + 100 , where x represents the number of items produced, in hundreds. The revenue, R, in thousands of dollars, that these items generate is given by the relation 𝑅 = 𝑥! − 10𝑥 + 250 , where x represents the number of items sold, in hundreds. a) Profit is defined as the difference between the revenue and the cost. Using P = R − C, develop a profit relation for the company. b) Determine the zeros of the profit relation. c) How many items should be produced to maximize profit?
d) What will the maximum profit be?
Answers
Section 1: Multiple Choice
1) a 2) c 3) a 4) c 5) d 6) d
Section 2: Completing the Square
2) a) vertex is (3, 8) and this is a max b) vertex is (-‐3, -‐14) and this is a min
Section 3: Solve By Factoring
3) a) -‐5 and 3 b) !! and !!
! c) !
! and 1
Section 4: Solve Using the Quadratic Formula
4) a) -‐4.29 and 0.62 b) no roots c) -‐0.81 and -‐6.19
Section 5: Write the Equation in Standard Form
5) factored form: 𝑦 = !!(𝑥 + 3)(𝑥 − 2) standard form: 𝑦 = 0.5𝑥! + 0.5𝑥 − 3
Section 6: Graph a Quadratic
6) a) -‐3 and 2.5 b) 𝑥 = −0.25 c) (-‐0.25, -‐15.1) d) see solutions page
Section 7: Application
7) a) 11.4 b) It reaches a max height of 3.5 m at a horizontal distance of 5.5 m.
Bonus Section
1) see your notes or the posted video for the proof 2) a) 𝑃 = −𝑥! + 19𝑥 + 150 b) -‐6 and 25 c) 950 d) $240 250
