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Math 121: Practice for Ch. 2 - Distance, Midpoint, Circles, Functions, Graphs, Exams of Algebra

Practice questions for chapter 2 of a college-level mathematics course, covering topics such as distance between points, midpoint of line segments, circles, functions, and their derivatives, as well as graphing. Students are asked to find distances, centers and radii of circles, slope-intercept forms of lines, and maximum and minimum values of quadratic functions.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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Math 121, Practice Questions for Chapter 2
1. (a) Find the distance between the points (โˆ’3,2) and (9,โˆ’3).
(b) find the midpoint of the line segment with endpoints (3,5) and (โˆ’5,2).
2. (a) Determine the center and radius of the circle whose equation is x2+y2+10x+4y+20 = 0.
(b) Write the equation of a circle whose center is (โˆ’5,1) and passes through the point (3,1).
(c) Find the equation of a circle that has diametral endpoints of (0,0) and (6,8). (Hint: the
center is the midpoint of the diametral endpoints).
3. Let f(x)=2x2+ 7 and g(x) = |xโˆ’1|, find
(a) (fโ—ฆg)(โˆ’5) (b) (gโ—ฆf)(x) (c) (fg)(0) (d) (f+g)(0)
4. Let f(x)=4x2โˆ’3x, find the difference quotient
f(x+h)โˆ’f(x)
h
5. Sketch the graph of f(x) = |x+ 3| โˆ’ 2 and find intervals where fis (a) increasing; (b)
decreasing. Is fone-to-one?
6. Determine the domains of the following functions.
(a) f(x) = x+ 3
(x+ 2)โˆš16 โˆ’x2.
(b) g(x) = โˆšxโˆ’4
(c) h(x) = โˆš4โˆ’x
(d) k(x) = 3(xโˆ’1)
(x+ 2)(xโˆ’11)
7. (a) Find the slope-intercept form of the line through the points (โˆ’1,3) and (4,โˆ’7).
(b) Find the slope-intercept form of the line that passes through the point (โˆ’3,โˆ’7) and is
perpendicular to the line 2x+ 5y= 10.
(c) Find the slope-intercept form of the line that passes through the point (โˆ’3,โˆ’7) and is
parallel to the line 2x+ 5y= 10.
8. (a) Write the quadratic function f(x) = โˆ’3x2+ 4xโˆ’5 in standard form by completing the
square. Using that information, sketch the graph of f(x).
(b) Find the vertex of the quadratic function f(x) = 3x2โˆ’6x+11, and find the range of f(x).
(c) Find the maximum of the quadratic function f(x) = โˆ’3x2+ 3x+7 and then find its range.
(d) Find the range of the quadratic funciton f(x) = x2โˆ’10x+ 3. Does this function have a
maximum or a minimum? If so, find it.
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Math 121, Practice Questions for Chapter 2

  1. (a) Find the distance between the points (โˆ’ 3 , 2) and (9, โˆ’3). (b) find the midpoint of the line segment with endpoints (3, 5) and (โˆ’ 5 , 2).
  2. (a) Determine the center and radius of the circle whose equation is x^2 +y^2 +10x+4y+20 = 0. (b) Write the equation of a circle whose center is (โˆ’ 5 , 1) and passes through the point (3, 1). (c) Find the equation of a circle that has diametral endpoints of (0, 0) and (6, 8). (Hint: the center is the midpoint of the diametral endpoints).
  3. Let f (x) = 2x^2 + 7 and g(x) = |x โˆ’ 1 |, find (a) (f โ—ฆ g)(โˆ’5) (b) (g โ—ฆ f )(x) (c) (f g)(0) (d) (f + g)(0)
  4. Let f (x) = 4x^2 โˆ’ 3 x, find the difference quotient

f (x + h) โˆ’ f (x) h

  1. Sketch the graph of f (x) = |x + 3| โˆ’ 2 and find intervals where f is (a) increasing; (b) decreasing. Is f one-to-one?
  2. Determine the domains of the following functions.

(a) f (x) = (^) (x + 2)x^ โˆš+ 3 16 โˆ’ x 2.

(b) g(x) = โˆšx โˆ’ 4 (c) h(x) = โˆš 4 โˆ’ x

(d) k(x) = (^) (x + 2)(3(x^ โˆ’x^ 1)โˆ’ 11)

  1. (a) Find the slope-intercept form of the line through the points (โˆ’ 1 , 3) and (4, โˆ’7). (b) Find the slope-intercept form of the line that passes through the point (โˆ’ 3 , โˆ’7) and is perpendicular to the line 2x + 5y = 10. (c) Find the slope-intercept form of the line that passes through the point (โˆ’ 3 , โˆ’7) and is parallel to the line 2x + 5y = 10.
  2. (a) Write the quadratic function f (x) = โˆ’ 3 x^2 + 4x โˆ’ 5 in standard form by completing the square. Using that information, sketch the graph of f (x). (b) Find the vertex of the quadratic function f (x) = 3x^2 โˆ’ 6 x + 11, and find the range of f (x). (c) Find the maximum of the quadratic function f (x) = โˆ’ 3 x^2 + 3x + 7 and then find its range. (d) Find the range of the quadratic funciton f (x) = x^2 โˆ’ 10 x + 3. Does this function have a maximum or a minimum? If so, find it.
  1. An air freight company has determined that its cost of delivering x parcels per flight is

C(x) = 875 + 0. 75 x

and it charges $12.00 per parcel to send each parcel. Find: (a) the revenue function; (b) the profit function; (c) the minimum number of parcels the company must ship on a flight to break even.

  1. The height in feet of a projectile with an initial velocity of 64 feet per second and an initial height of 80 feet is a function of time t in seconds, given by

h(t) = โˆ’ 16 t^2 + 64t + 80.

(a) Find the maximum height of the projectile. (b) Find the time t when the projectile reaches its maximum height. (c) Find the time t when the projectile hits the ground (has a height of 0 feet).

(d) The difference quotient h(1 1 .01). 01 โˆ’โˆ’^ h. 99 (.99) gives the average velocity of the projectile for

. 99 โ‰ค t โ‰ค 1 .01. Compute this difference quotient. Do you think it would provide a good estimate of the instantaneous velocity of the projectile when t = 1?

  1. (a) Do # 41, p. 275. (b) Determine whether the graph of y = x^3 โˆ’ 4 x is symmetric about the (i) x-axis, (ii) y-axis, (iii) origin. (c) Determine whether the function g(x) = x^5 โˆ’ x^3 is even, odd or neither. (d) In terms of shifts or translations, how does the graph of y = f (x + 5) โˆ’ 10 compare to the graph of y = f (x)? (e) In terms of shifts or translations, how does the graph of y = f (x + 5) โˆ’ 10 compare to the graph of y = f (x โˆ’ 3) + 2?
  2. Find two numbers whose difference is 10 and the sum of whose squares is a minimum.
  3. Let f (x) = โˆš 5 โˆ’ x and g(x) = โˆšx + 7. Find the domain of (i) f + g, (ii) f โˆ’ g, (iii) f g, (iv) f /g.
  4. A farmer has $1000 to spend to fence a rectangular corral. Because extra reinforcement is needed on one side, the corral costs $6 per foot along that side. It costs $2 per foot to fence the remaining sides. What dimensions of the corral will maximize the area of the corral?
  5. A Hollywood charter bus company that provides tours through the movie star neighbor- hoods in Beverly Hills has determined that the cost of providing x people a tour is

C(x) = 180 + 2. 50 x