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Math 100 โ Exam 2 Review Worksheet Fall 2013
Instructor: Cheryl Jaeger Balm
Important: This is in no way meant to encompass all the material covered by Exam 2. The purpose of this worksheet is to give you extra practice on the types of problems that you said you were most concerned about.
1 Problems of the form y = a(x โ h)^2 + k
Example: Put y = 3x^2 โ 9 x + 10 into the form y = a(x โ h)^2 + k. Method 1: Completing the square
y = 3x^2 โ 9 x + 10
= 3
x^2 โ 3 x +
Factor out a
x^2 โ 3 x +
Add and subtract
b
. Here b = 3.
= 3
x^2 โ 3 x +
Add parentheses
x โ
x โ
x โ
y = 3
x โ
Distribute a
Method 2
Remember h = โb 2a and a is the same in y = ax^2 + bx + c and y = a(x โ h)^2 + k. y = 3x^2 โ 9 x + 10, so a = 3, b = โ 9 , c = 10. h =
(h, k) is a point on the parabola (the vertex!) so plug h into the original equation to get k:
y = 3x^2 โ 9 x + 10
k = 3
k =
a = 3, h =
, k =
so y = 3
x โ
Practice problems
- Put y = x^2 + 4x + 1 into the form y = a(x โ h)^2 + k.
- Put y = โx^2 + 12x โ 5 into the form y = a(x โ h)^2 + k.
- Put y = โ 2 x^2 + 8x โ 5 into the form y = a(x โ h)^2 + k by completing the square.
3 Quadratic equations and inequalities
Example - Equation: Solve x^2 โ x โ 6 = 0. Solving a quadratic equation is the same as finding the roots. You can factor, use completing the square, quadratic formula, etc.
x^2 โ x โ 6 = 0 (x โ 3)(x + 2) = 0 x = 3 or โ 2
Example - Inequality: Solve x^2 โ x โ 6 < โ4.
x^2 โ x โ 6 < โ 4 x^2 โ x โ 2 < 0 Move everything to one side.
This is the quadratic inequality we care about from now on! Set it equal to 0 and find the roots using any method.
x^2 โ x โ 2 = 0 (x โ 2)(x + 1) = 0 x = 2 or โ 1
Method 1: Graphing Since 2 and โ1 are the roots of y = x^2 โ x โ 2, (2, 0) and (โ 1 , 0) are two points on this parabola. We need one more point, so we can plug in x = 0 (or any other number that is not a root). y = x^2 โ x โ 2 = 0^2 โ 0 โ 2 = โ2 so (0, โ2) is another point on our parabola.
x
y y = x^2 โ x โ 2
When is x^2 โ x โ 2 < 0? When โ 1 < x < 2.
Method 2: Number line x^2 โ x โ 2 = 0 at x = โ1 and x = 2, so we need to plug in a number less than โ1, a number between โ1 and 2, and a number greater than 2 into f (x) = x^2 โ x โ 2.
f (โ2) = (โ2)^2 โ (โ2) โ 2 = 4 + 2 โ 2 = 4 > 0 f (0) = 0 โ 0 โ 2 = โ 2 < 0 f (3) = 3^2 โ 3 โ 2 = 9 โ 3 โ 2 = 4 > 0
f (x)
x โ 1
pos neg 0 pos
When is f (x) = x^2 โ x โ 2 < 0? When โ 1 < x < 2.
Practice problems
- Solve y = โ 2 x^2 + 4x + 6.
- Solve
x โ 1 < x + 1
4 Piecewise functions
- Given the following function f (x), find f (0), f (1), f (2), and f (5).
f (x) =
2 x + 1, if x > 5 x โ 5 , if 1 < x โค 5 0 , if x โค 1 .
- Given the following function g(x), find g(โ2), g(0), g(2), and g(4).
g(x) =
7 x, if x < โ 2 x^2 , if โ 2 โค x โค 2 1 2 x^ + 1,^ if^ x >^2 .
- Suppose you are biking to a friendโs house. For the first 2 minutes you are riding, you go 8 MPH. You then have a wait at a stop sign for 1 minute for traffic to clear. After that, you bike at 10 MPH for 3 more minutes until you arrive at your friendโs house. Write a piecewise function v(t) which expresses your velocity v in MPH in terms of t, the number of minutes since you left your house. Then graph v(t).
5 Combining functions
- If f (x) = x^2 โ 1 and g(x) = 2 + 7x, find each of the following. Be sure to simplify your answers. - f (x)g(x) - f (g(x)) - g(f (x))
- If A(x) = x โ 2 x^2 and B(x) = 3x โ 7, find A(B(2)) and B(A(2))
