Quadratic Functions and Equations
Unit Review
1.
Determine the coordinates of the vertex, axis of symmetry, maximum or minimum value, domain, and range
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for the function, 𝑦 = − 2 (𝑥 − 3) + 5.
2.
Determine a quadratic function in vertex form for the parabola.
3.
A farmer has 200 m of fencing material to enclose a rectangular field adjacent to a river. No fencing is required
along the river.
a) Write a function that can be used to represent the area of the field.
b) Determine the maximum area of the field.
c) Determine the dimensions of the region that give the maximum area.
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4.
Convert the function y = 2x2 + 30x + 117 to the form y = a(x – p)2 + q. State the coordinates of the vertex, axis
of symmetry, maximum or minimum value, domain, and range.
5.
If a farmer harvests his crop today, he will have 1200 bushels worth $6 per bushel. Every week he waits, the
crop yield increases by 100 bushels, but the price drops $0.30 per bushel.
a) What quadratic function can be used to model this situation?
b) When should the farmer harvest his crop to maximize his revenue? What is the maximum revenue?
6.
Solve by graphing t 2 – 5t – 150 = 0.
7.
Determine the real roots to the quadratic equation 6x2 + 2x – 4 = 0 by factoring. Show work.
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8.
Two numbers have a sum of 22. What are the numbers if their product is 96?
9.
Determine the real roots of each quadratic equation. Express your answers as exact values.
a) x2 + 4x – 1 = 0
b)
4x2 – 4x – 7 = 0
10.
Determine the real roots of each equation algebraically. Choose a different method for each equation, and explain
why you chose that method. Express your answers as exact values in simplest form.
a)
x2 – 10x + 16 = 0
b)
3x2 + 19x – 14 = 0
c)
x2 – 6x + 7 = 0
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Solutions:
1. 𝑦 = −2(𝑥 − 5)2
2. a) 𝐴(𝑤) = −2𝑤 2 + 200𝑤 b) Max Area: 5000 m2
c) l = 100m w = 50 m
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3. 𝑦 = 2(𝑥 + 7.5) + 4.5
4a) 𝑅(𝑥) = −30𝑥 + 240𝑥 + 7200 OR 𝑅(𝑥) = −30(𝑥 − 4)2 + 7680
4b) He should wait 4 weeks to maximize revenue to $7680.
7. 6 and 16
8a) 𝑥 = −2 ± √5
5. 𝑡 = −10, 𝑡 = 15
8b) 𝑥 =
9b) Quadratic formula or Factor by decomposition: 𝑥 = −7, 2⁄3
1±2√2
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9a) Easily factorable: 𝑥 = 8, 𝑥 = 2
9c) Quadratic formula only 𝑥 = 3 ± √2
9d) Square root principal/ Solve for x. Only one variable, so solve for it. 𝑥 = 1, 𝑥 = 5
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6. 𝑥 = −1, 𝑥 = 3