⏱️ Duration: 75 minutes | Total: /60 marks Show all work. Answers without supporting work receive partial credit at best.
K/U
/15
Thinking
/15
Comm.
/15
Applic.
/15
Part A: Knowledge & Understanding [15 marks]
Question 1 [3 marks]
Factor completely: \( 3x^2 - 12x - 15 \).
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Question 2 [2 marks]
The vertex of \( y = -3(x + 2)^2 + 7 \) is:
Solution:
From \( y = a(x - h)^2 + k \): \( h = -2 \), \( k = 7 \).
Question 3 [3 marks]
Convert \( y = 2x^2 - 12x + 5 \) to vertex form by completing the square.
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Question 4 [2 marks]
For \( 2x^2 - 4x + 5 = 0 \), the discriminant is:
Solution:
\( \Delta = (-4)^2 - 4(2)(5) = 16 - 40 = -24 \). Negative → no real roots.
Question 5 [3 marks]
Solve \( x^2 - 8x + 12 = 0 \) by factoring.
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Question 6 [2 marks]
Solve \( 2x^2 - 5x - 3 = 0 \) using the quadratic formula. The positive root is:
Solution:
\( x = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm 7}{4} \) → \( x = 3 \) or \( x = -\frac{1}{2} \).
Part B: Thinking & Inquiry [15 marks]
Question 7 [5 marks]
Determine the value of \( k \) so that \( x^2 + kx + 9 = 0 \) has exactly one real root. Justify all reasoning.
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Question 8 [5 marks]
A quadratic function passes through \( (1, 0) \), \( (5, 0) \), and \( (3, -8) \). Determine the equation in standard form. Show all work.
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Question 9 [5 marks]
Two numbers differ by 6 and their product is 91. Find both numbers using a quadratic equation. Verify your answer.
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Part C: Communication [15 marks]
Question 10 [4 marks]
Explain how the discriminant determines the number of real roots of a quadratic equation. Provide a numerical example for each case (\( \Delta > 0, = 0, < 0 \)).
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Question 11 [3 marks]
Compare and contrast the three forms of a quadratic function (standard, vertex, factored). State which features are immediately apparent in each form.
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Question 12 [4 marks]
Describe step by step how to convert \( y = ax^2 + bx + c \) to vertex form by completing the square. Use \( y = 3x^2 + 12x - 5 \) as your worked example.
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Question 13 [4 marks]
A student claims, "Every parabola has two x-intercepts because the quadratic formula always gives two answers." Critique this claim with mathematical evidence.
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Part D: Application [15 marks]
Question 14 [5 marks]
A toy rocket is launched from a 2 m platform with initial speed 30 m/s. Its height is \( h(t) = -5t^2 + 30t + 2 \). Find: (a) the maximum height; (b) the time it returns to the ground; (c) the time(s) the rocket is at 32 m.
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Question 15 [5 marks]
A farmer has 60 m of fencing to enclose a rectangular pen against a barn (the barn is one full side, so no fencing needed there). Determine the dimensions that maximize area, and state the maximum area.
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Question 16 [5 marks]
A company's revenue is \( R(x) = -2x^2 + 80x \) (in thousands of dollars), where \( x \) is the price in dollars. (a) Find the price that maximizes revenue. (b) Find the maximum revenue. (c) For what range of prices is revenue at least \$600 thousand?
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Evaluation Rubric
Level
Description
%
4
Thorough, insightful, high degree of effectiveness