MATH 1314 - College Algebra
Review for Test 2
Sections 3.1 and 3.2
1. For f (x) = −x 2 + 4 x + 5 , give (a) the x-intercept(s), (b) the y-intercept, (c) both coordinates of the
vertex, and (d) the equation of the axis of symmetry. (e) Graph f (x) .
2
€ 2. For f (x) = 2(x − 3) − 8 , give (a) the x-intercept(s), (b) the y-intercept, (c) both coordinates of the
vertex, and (d) the equation of the axis of symmetry. €
(e) Graph f (x) .
€
3. The graph of the quadratic function f (x) is shown at the
right. Determine the function's formula in the form
€
f (x) = a(x − h) 2 + k .
(–1,8)
8
6
€
€
y
4
–4
–2 –1
–3
2
1
x
–2
–4
–6
4. The graph of the quadratic function f (x) is shown at the
right. Determine the function's formula in the form
6
4
f ( x) = ax 2 + bx + c .
€
2
–1 –2
€
y
1
2
3
–4
–6
–8
(1,–8)
5. A baseball is hit so that its height s in feet after t seconds is given by s(t) = −15t 2 + 40t + 5 .
(a) Find the maximum height of the baseball. (b) When does the baseball hit the ground?
6. Determine the zeros of the following quadratic functions and simplify your answers:
€
(a) f ( x) = 1 x 2 − 3 x +1
(b) f ( x) = 2x 2 − 6x +1
4
€
2
7. Solve x 2 − 8x − 4 = 0 by completing the square. Simplify your answer.
€
€
4
x
MATH 1314 - College Algebra - Review for Test 2 (Thomason) - p. 2 of 8
2x − 3
8. Determine the domain of f (x) = 2
. Give your answer in set-builder notation.
x − 3x −10
9. For 3x 2 − 5x + 2 = 0 , (a) calculate the discriminant and (b) give the number of real solutions.
Section 3.3
€
€
10. For each of the following give your answer in a + bi form.
(a) Add: (2 + 3i) + (−5 − 4i)
(c) Multiply: (3 − 2i)(−4 + i)
€
(b) Subtract: (−5 + 4i) − (3 − 2i)
4 + 3i
(d) Divide:
5 − 2i
€
€
11. Solve 3x 2 = 6x − 4 and simplify your answer. (Simplify radicals, reduce fractions, and express
any €
imaginary numbers in terms of i.)
€
12. How many real zeros does each of the following functions have?
€
Section 3.4
13. The graph of f (x) = −x 2 − 2x + 3 is shown at the right.
Solve f (x) ≤ 0 and give your answer in interval notation.
(–1,4)
€
€
(0,3)
(1,0)
(–3,0)
14. For the function f ( x) whose graph is show in Problem 13, for what values of x is f ( x)
(a) increasing and (b) decreasing? Give your answers in interval notation.
€
€
MATH 1314 - College Algebra - Review for Test 2 (Thomason) - p. 3 of 8
15. Solve the inequality and write the solution set in interval notation: 7x 2 − 9x > 0
16. Solve the following inequality and give your answer in interval notation: x 2 + x ≤ 12
€
Section 3.5
€
17. The graph of y = f (x) is shown on the coordinate
system at the right. Sketch the graphs of
(a) y = f (x + 2) −1,
(b) y = f ( x −1) + 2 ,
€
(c) y = − f 1 x , and
y
( )
1
2
€
€
(d) y = 2 f (−x) .
1
x
€
€
18. Let f (x) = x . Write a formula for a function g whose graph is similar to f (x) but is shifted
right 3 units and up 4 units.
€
Section 4.1
€
19. The graph of f (x) is shown on the coordinate system
at the right. Determine the (a) local minima, (b) local
maxima, (c) absolute minimum, (d) absolute maximum,
(e) intervals in which f (x) is increasing, and (f) intervals
€
in which f (x) is decreasing, if any. Give your answers to
parts (e) and (f) in interval notation.
€
3
2
1
–3
–2
–1
1
2
3
–1
€
–2
–3
–4
20. Determine whether each of the following is an even function, an odd function, or neither. Show or
explain how you determined your answer.
(a) f (x) = x 2 − 3
(b) f (x) = (x − 3) 2
(c) f (x) = x 2 + x
€
21. The table at the right is a complete
representation of f. Is f an even function, an odd x
€ or explain how you f(x)
€
function, or neither? Show
determined.
–5
8
–3
4
–1
2
0
0
1
2
3
4
5
8
MATH 1314 - College Algebra - Review for Test 2 (Thomason) - p. 4 of 8
Section 4.2
22. Use the graph of the polynomial function f (x) shown
at the right to answer the following.
(a) How many turning points does the graph have?
€
y
1
(b) Estimate the x-intercepts, assuming they are integers.
1
x
(c) Is the leading coefficient of f (x) positive or is it
negative?
(d) What is the minimum degree of f (x) ?
€
23. For f ( x) = −2x 3 + 5x 2 €
+14x − 35 , (a) give the degree, (b) give the leading coefficient, (c) state
the end behavior as x →−∞ , and (d) state the end behavior as x →∞ .
4
2
€ 24. For f ( x) = −x + 2x − 8 , (a) give the degree, (b) give the leading coefficient, (c) state the end
behavior as
€ x →−∞ , and (d) state the end behavior as
€ x →∞ .
25. Sketch a graph of a polynomial that satisfies the following conditions: Degree 3 with two real
€ zeros and a negative leading coefficient
€
€
26. Sketch a graph of a polynomial that satisfies the following conditions: Degree 4 with three real
zeros and a negative leading coefficient
⎧4
for x < −2
⎪ 2
27. Let f ( x) = ⎨ x
for − 2 ≤ x ≤ 3 .
⎪
⎩ x − 3 for x > 3
€
(a) Determine f (−3) . (b) Determine f (3) . (c) Graph f ( x) . (d) Give any values of x at which
f ( x) is not continuous.
Section 4.3
€
€
€
€
28. Divide x 3 − 4 x 2 − 20x − 3 by x + 3.
29. Divide 6x 3 − x 2 + 4 x − 7 by 3x − 2 .
€
€
30. Is x + 2 a factor of f (x) = x 3 + 5x 2 + 3x − 6 ? Tell how you determined your answer.
€
€
31. What is the remainder when x 3 − 3x 2 + 4 x − 5 is divided by x − 2 ?
€
€
€
€
MATH 1314 - College Algebra - Review for Test 2 (Thomason) - p. 5 of 8
Section 4.4
32. The graph of a 3rd, 4th, or 5th degree polynomial f (x) with
integer zeros is shown at the right. Determine the factored
form of f (x) .
€
€
33. Solve exactly for x: 7x 3 − 5x 2 − 21x + 15 = 0 . Simplify your answers including removing
perfect squares from under square roots and reducing fractions, when possible.
2
34. Solve exactly
€ for x: x = 4 x −13. Simplify your answers including removing perfect squares
from under square roots and reducing fractions, when possible. Write any complex solutions in
standard form.
€
35. Solve exactly for x: x 6 = 10x 3 − 21. Simplify your answers including removing perfect squares
from under square roots and reducing fractions, when possible. Write any complex solutions in
standard form.
€
2
36. Find the zeros of f (x) = 7x 3 + 5x 2 + 12x − 4 given that is a zero.
7
Section 4.5
€
37. The graph of a 5th degree polynomial f (x) is€shown at
the right.
(a) How many different real zeros does f (x) have?
€
y
3
(b) How many different imaginary zeros does f (x) have?
€
3
€
38. Find the completely factored form of a polynomial f (x) with real coefficients that satisfies the
following conditions: Degree 3; a3 = 2 ; zeros include 3 and 1–2i
€
€
x
MATH 1314 - College Algebra - Review for Test 2 (Thomason) - p. 6 of 8
39. Find the completely factored form of a polynomial f (x) with real coefficients that satisfies the
following conditions: Degree 4; an = −1; zeros include 0, 2, and 3i
40. (a) Find all the zeros of f ( x) = x 3 − 3x 2 +€9x − 27 . (b) Write f (x) in completely factored form.
€
41. (a) Find all the zeros of f ( x) = 4x 3 + 32x . (b) Write f (x) in completely factored form.
€
€
4
3
2
42. (a) Find all the zeros of f ( x) = 3x − 6x +12x . (b) Write f (x) in completely factored form.
€
€
Section 4.7 (Polynomial Inequalities )
€
€
43. Solve for x and give your solution in interval notation: 2x 2 +15x < x 3
44. Solve for x and give your solution in interval notation: (x + 4)(x − 2)(x − 5) 2 ≤ 0
Answers
1. (a) (−1,0), (5,0)
(b) (0,5)
€ (c) (2,9)
(d) x = 2
€
2. (a) (1,0), (5,0)
(b) (0,10)
(c) (3,–8)
(d) x = 3
€
€
€
MATH 1314 - College Algebra - Review for Test 2 (Thomason) - p. 7 of 8
3. f ( x) = −2( x +1)2 + 8
6. (a) 3 ± 5 (b)
€
4. f ( x) = 2x 2 − 4x − 6
3± 7
2€
6i
(b) −8 +€
€ (c) −10 +11i
€
(d)
€
14 23
+ i
29 29
11. 1 ±
€
€
17
(a)
y
€
1
1
x
17
(b)
1
1
x
y
1
x
y
13. (−∞, −3][1, ∞)
€
(b) [−1, ∞)
€
€
€
9. (a) 1 (b) 2
€
3
i
3
14. (a) (−∞, −1]
€
}
8. x x ≠ −2,5
€
y
12. (a) 2 (b) 0 (c) 1
€
{
7. 4 ± 2 5
10. (a) −3 − i
€
2
4 + 19
5. (a) 31 ft (b)
sec
3
3
17
(c)
1
1
x
17
(d)
1
⎛ 9 ⎞
15. (−∞,0)⎜ , ∞⎟
⎝ 7 ⎠
16. [−4,3]
18. g( x) = f ( x − 3) + 4 = x − 3 + 4
19. (a) 0, –4 (b) 1 (c) -4 (d) none (e) [−2, −1], [1, ∞) (f) (−∞, −2], [−1,1]
€
20. (a) Even, because the graph is an open up parabola that is symmetric about the y-axis.
2
2
2
2
(b) Neither, because f (−x) = (−x
9 but f ( x)
€ − 3) =€x + 6x + €
€= ( x − 3) = x − 6x + 9 so
f (−x) ≠ f ( x) and f (−x) ≠ − f ( x) .
€
€
(c) Neither, because f (−x) = (−x)2 + (−x) = x 2 − x but f ( x) = x 2 + x so f (−x) ≠ f ( x) and
f (−x) ≠ −€
f ( x) .
€
€
21. Even, because f (−x) = f ( x) for all values of x in the domain of f.
€
€
€
22. (a) 3 (b) –3, –1, 2 (c) positive (d) 4th
€
MATH 1314 - College Algebra - Review for Test 2 (Thomason) - p. 8 of 8
23. (a) 3rd (b) –2 (c) f ( x) →∞ (d) f ( x) →−∞
24. (a) 4th (b) –1 (c) f ( x) →−∞ (d) f ( x) →−∞
€
25. Answers may vary.
€
€
26. Answers may vary.
€
6
28. x 2 − 7x +1 −
x +3
27. (a) 4 (b) 9
29. 2x 2 + x + 2 −
€
(c)
€
3
3x − 2
30. Yes, because the remainder given by
f (2) or division is 0.
31. −1
€
(d) 3
€
1
32. f ( x) = ( x +1)2 ( x − 2)( x − 4)
2
(d)
33. ± 3 ,
5
7
34. 2 ± 3i
2
1
7
36. €, − ±
i
7
2
2
35. 3 3 , 3 7
€
38. f ( x) = 2( x − 3)[x − (1 − 2i)][x − (1+ 2i)]
€
€ €
€
€
39.
f ( x) = −x( x − 2)( x − 3i)( x + 3i)
€
40. (a) 3, ±3i (b) f ( x) = ( x − 3)( x − 3i)( x + 3i)
€
41. (a) 0, ±2i 2 (b) f ( x) = 4x( x − 2i 2 )( x + 2i 2 )
€
⎡
⎤⎡
⎤
42. (a) 0, 1 ± i 3 (b) f ( x) = 3x 2 ⎢ x − 1 − i 3 ⎥⎢ x − 1+ i 3 ⎥
⎣
⎦⎣
⎦
€
€
43. (−3,0)(5, ∞)
44. [−4,2]{5}
€
€
(
€
€
)
(
)
37. (a) 2 (b) 2