MATH 111
Departmental Midterm Exam Review
Exam date: Tuesday, March 1st
Exam will cover sections 2.4-2.9 + 3.1-3.3 and will be NON-CALCULATOR EXAM.
1. Terms to know: quadratic function, axis of symmetry, vertex, minimum/maximum value,
increasing/decreasing piece-wise defined function, continuous and discontinuous function,
absolute value function, arithmetic combinations of function, function composition, inverse
function, symmetry, function transformations, domain and range of the function, difference
quotient, derivative of the function, polynomial function, degree, leading coefficient, zeros, roots,
factors, complete factorization, long division, synthetic division, complex number, real and
imaginary parts, complex conjugate,
2. Be able to use the completion of squares to rewrite quadratic function in the vertex form
f ( x ) = a ( x − h) 2 + k .
3. Be able to find the x- and y-intercepts, axis of symmetry, vertex, minimum/maximum value,
increasing/decreasing intervals, domain and range and sketch the graph of the quadratic function
4. Be able to use the quadratic formula to factor and solve the quadratic equation.
5. Know how to use the discriminant to determine the number of real solutions of the given quadratic
equation.
6. Know and be able to use vertical and horizontal shifts, vertical compression and elongation of the
graph and symmetry to graph quadratic functions.
7. Know the applications of quadratic functions.
8. Be able to graph piece-wise defined functions and absolute value functions.
9. Know the arithmetic combinations of functions, be able to find them and give the domain for each
combination.
10. Be able to find composition of two and three functions and give the domain for each composition.
11. Be able to write the function as the composition of few functions.
12. Be able to find the inverse function.
13. Be able to find the domain and range of the inverse function.
f ( x + h) − f ( x )
14. Be able to find and simplify difference quotient
.
h
15. Be able to use the definition of the derivative to find the derivative of a quadratic, cubic, rational
and square root functions.
16. Be able to write an equation of the function in terms of one variable based on the given information.
17. Be able to build a function from words.
18. Be able to find the equation of the polynomial function from the given graph, and from the given
conditions (zeros, intercepts, points, end behavior).
19. Be able to sketch the graph of the polynomial function based on the given conditions (zeros,
multiplicity, end behavior, etc).
20. Know the Division Algorithm.
21. Be able to divide polynomial by another polynomial using long division and synthetic division.
22. Theorem to know: Intermediate Value Theorem, Factor Theorem, Remainder Theorem, Fundamental
Theorem of Algebra, Complete Factorization Theorem, Conjugate Zeros Theorem, Rational Zeros Theorem.
23. Be able to find all exact zeros (rational, irrational and complex) of a function both algebraically and
graphically.
24. Be able to recognize when a function has complex roots both algebraically and graphically, and find all
exact complex roots of the polynomial.
25. Be able to give complete factorization of the polynomial function of degree 3, 4 and 5.
26. Be able to verify if the indicated complex number is a zero of the polynomial function and use
Conjugate Zeros Theorem to find ALL zeros of the polynomial function.
Partial Review Exercises
1. Write the function in the form f ( x) = a ( x − h) 2 + k : a). f ( x) = 2 x 2 − 12 x + 11 ; b) y = − 12 x 2 + x + 52 ;
c) f ( x) = − x 2 + 8 x − 7
2. Given f ( x) = x and g ( x) = 3 5 − x . Find a). f  g; b). g  f and give the domain for each composition.
3. Solve the given equation: a) x 2 + 8 x − 1 = 0 ; b) 9 + 13x − 10 x 2 = 0 .
4. If a projectile is shot vertically upward from the ground with an initial velocity of 100 feet per
second, neglecting air resistance, its height s (in feet) above the ground t seconds after projection can
be modeled by s = −16t 2 + 100t .
a) How long will it take for the projectile to return to the ground?
b) When will it reach maximum height?
c) What is the maximum height?
5. Sketch the graph
 x, x ≥ 0
a). f ( x) = 
− 2, x < −1
x 2 , x < 1
b). f ( x) = 
c) y = x 2 + x − 6 .
 x + 1, x > 1
6. An open rectangular box with a volume of 12 ft 3 has a square base. Express the surface area A of the
box as a function of the length x of the side of the base.
7. Find the number of units that produce the maximum revenue, R = 900 x − 0.1x 2 , where R is the total
revenue in dollars and x is the number of units sold.
8. Given f ( x) = x − 7 , g ( x) = x 2 , h( x) = x 3 + 6 . Find f  g  h .
9. Given f(x)= 3 ; g(x)= 2 − x .
x+1
Find a). f  g; b). g  f and give the domain for each composition.
10. Given f(x) = x 2 ; g(x) = x - 3.
Find a). (f  g)(5); b). (g  f)(7); c). (f  f)(2).
11. Given f(x) = x and g(x) = 4 − x 2 .
Find a). ( gf ) (x); b). (fg)(x) and give the domain for each combination.
f ( x + h) − f ( x )
for the following functions: a). 𝑓(𝑥) = 7𝑥 − 3; b). f ( x) = x 2 + 2 x ;
h
2
c). f ( x) =
; d). f ( x) = x + 3 .
x−5
13. A rectangle has perimeter 60 meters. Express the area A of the rectangle as a function of
the length x of one of its sides.
12. Find
14. A rectangle has an area of 36 square meters. Express the perimeter P of the rectangle as
a function of the length x of one of its sides.
15. A farmer wants to enclose a rectangular field by a fence and divide it into two smaller rectangular
fields by constructing another fence parallel to one side of the field. He has 3000 yards of fencing.
Find the dimensions of the field so that the total area is a maximum.
16. Express the area A of a circle as a function of its circumference C.
3
17. Find the derivative of a) f ( x) = 3 x 2 − 7 x ; b) f ( x) =
.
x+5
18. An open box is to be made from a square piece of material 32 centimeters on a side by cutting
equal squares (x centimeters) from the corners and turning up the sides. Determine the volume of the box
as the function of x.
19. Find the inverse function
a). f ( x) =
2x + 5
; b). f(x) = 2x – 5; c). y = 2 +
x −1
x − 3 ; d). y = ( x − 2) 2 , x ≥ 2 ; e) f ( x) =
x
.
x+3
20. A closed rectangular box with a square base x is and height h to be constructed from 300 square
inches of material. Express the volume of the box as a function of the length x of the side of the base.
21. Use the table below to find the following values
x
f(x) g(x)
0
4
4
1
2
3
2
0
2
3
1
1
4
4
0
Find a). (fg)(2); b). (gf)(2); c) (f + g) (3).
22. Find the maximum or the minimum value of f ( x) = −2 x 2 − 10 x + 5 .
23. Find the largest interval on which f ( x) = −( x + 1) 2 is increasing.
24. Express the distance d , from a point ( x, y ) on the graph of x + y = 3 to the point (5, 8) as a function of x .
25. Determine the equation of the quadratic function whose graph is shown below.
3
2
1
-3
-2
-1
1
2
3
-1
-2
-3
-4
26. An arrow is shot vertically upward with an initial velocity of 64 ft/s from a point 6 ft above
the ground. The height of an arrow above ground is described by s (t ) = −16t 2 + 64t + 6 .
a) What is the maximum height attained by the arrow?
b) At what time does the arrow fall back to the 6-ft level?
27. A rectangular plot of land will be fenced into three equal portions by two dividing fences
parallel to two sides. If the total fence to be used is 400 m, find the dimensions of the land that
has the greatest area.
28. Find an equation of the polynomial function below that passes through (2,-6)
29. A rectangular plot of land will be fenced into two equal portions by one dividing fence
parallel to one side. If the area to be enclosed is 1,000 m 2 , express the amount of fence required
as a function of the width x of the rectangular plot.
30. a). If 𝑥 = 2 and x = −2 are zeros of f ( x) = x 4 + 2 x 3 − 2 x 2 − 8 x − 8 , find all zeros of the polynomial.
b). If 𝑥 = 2i is a zero of f ( x) = x 4 + 2 x 2 − 8 , find all zeros of the polynomial.
c). If 𝑥 = 1 is a zero of f ( x) = x 3 + x 2 − 3 x + 1 , find all zeros of the polynomial.
1
d) If 𝑥 = − 2 is a zero of f ( x) = 2 x 3 − x 2 + x + 1 , find all zeros of the polynomial.
31. Use the Remainder Theorem to find r when f ( x) = 4 x 3 − x 2 + 4 is divided by x − 2 .
32. If 𝑥 = 2/3 is a zero of the polynomial f ( x) = 3x 3 − 8 x 2 − 8 x + 8 , find the complete factorization of 𝑓(𝑥).
33. Write each expression in standard complex form: a + bi a)
8−i
; b)
2+i
− 8 + − 18 .
34. Find a polynomial of lowest degree having only real coefficients and zeros at 3 and 2 + i. Write
your answer in expended form.
35. Find a polynomial of lowest degree having only real coefficients and zeros at x = 1+2i and x = 2
of multiplicity 2. Write your answer in expended form.
36. Determine the end behavior of P( x) = − x 4 − 5 x 2 + 8 x + 8 ; y → a as x → ∞ and y → b as x → −∞ .
37. If x = 12 i is a zero of the polynomial f ( x) = 12 x 3 + 8 x 2 + 3 x + 2 , find all zeros.
38. Sketch the graph of polynomial P(x) that has zeros of multiplicity one at x = 0 and x = 1, has a
zero of multiplicity three at x = -3, and satisfies P(x) → −∞ as x → −∞ and P(x) → ∞ as x → ∞ .
39. Graph a) f ( x) = 2 x 4 − 8 x 3 ; b) f ( x) = − x 3 + x 2 + 2 x ; c) p( x) = x 4 − 16 x 2 .
Answers:
1. a). 2( x − 3) 2 − 7 ; b). − 12 ( x − 1) 2 + 3 ; c) − ( x − 4) 2 + 9 .
2. a) 6 5 − x , D (−∞,5] ; b) 3 5 − x , D [0, ∞) .
3. a) x = −4 ± 17 ; b) x = −1 / 2; x = 9 / 5 .
4. a). 6.25 s., b) 25/8 s; c) 625/4 ft.
5.
a)
b)
c)
5
5
4
4
3
3
2
2
5
1
1
4
7
6
3
-5
-4
-2
-3
-1
1
2
3
4
-5
5
-4
-3
-2
-1
1
2
3
4
5
2
-1
-1
-2
-2
-3
-3
-4
-4
-2
-5
-5
-3
1
-5
-4
x
7. 4500 units
( x 3 + 6) 2 − 7 .
9. a).
3
;
2 − x +1
10. a). 4; b). 46; c). 16.
x
11. a).
; D: [0, 2) . b).
4 − x2
12. (a). 7 ;
2 x −1 ;
x +1
D: (−∞,2] ; b).
D: (−∞, −1) ∪ [1/ 2, ∞) ;
4x − x3 ; D: [0, 2].
(b). 2x + h +2; (c).
−2
; (d).
( x + h − 5)( x − 5)
1
x+h+3 + x+3
.
13. A(x) = x(30 - x)
14. P(x) = 2x + 72/x
15. 500 by 750
16. A =
17. a) 6 x − 7; b)
−3
( x + 5) 2
.
18. V(x) = x (32 − 2 x ) 2 ;
19. a). f −1 ( x) =
3
5+ x
; b).
x−2
20. V = 75 x − x2 .
21. a). 0, b). 4, c). 2.
22. 35/2.
23. (−∞,−1] .
x +5
2
; c). ( x − 2) 2 + 3, x ≥ 2 ; d).
-2
-1
1
-1
6. A =
x 2 + 48 , x > 0 .
8.
-3
x + 2 ; e) f −1 ( x) = 13−xx .
2
3
4
5
24. d = 2 x 2 + 50 .
25. y = 2( x − 1) 2 − 3 .
26. a) 70ft; b) 4 s.
27. 50 m by 100 m.
28. y = 15 x 5 − 2 x 3 + 95 .
29. P = 3x +
2, 000
x
.
30. a). x = -2, x = 2, x = −1 ± i ; b). x = ± 2 ; x = ±2i ; c). x = 1; x = − 1 ± 2 ,
d) x = − 12 , x = 12 ± 23 i ; e) x = 1, x = −3 ± 7 .
31. 32.
32. 3( x − 23 )( x − 1 − 5 )( x − 1 + 5 )
33. a) 3-2i; b) 5 2i .
3
2
34. x − 7 x + 17 x − 15
4
3
2
35. x − 6 x + 17 x − 28 x + 20
36. a = −∞, b = −∞ .
37.
38.
x = − 23 ; x = ± 12 i
.
39 a)
b)
39.
40.
41.
42.
43.
44.
c)