MATH 1314 - College Algebra Review for Test 2

Sections 3.1 and 3.2 1. For

€

f (x) = −x2 + 4x + 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. For

€

f (x) = 2(x − 3)2 − 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 .

4. The graph of the quadratic function

€

f (x) is shown at the right. Determine the function's formula in the form

€

f ( x) = ax2 + bx + c .

5. A baseball is hit so that its height s in feet after t seconds is given by

€

s(t) = −15t2 + 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) = 14 x2 − 3

2 x +1 (b)

€

f ( x) = 2x2 − 6x +1 7. Solve

€

x2 − 8x − 4 = 0 by completing the square. Simplify your answer.

y

x–4 –3 –2 –1 1

–2–4–6

24

68

(–1,8)

1 2 3 4–1

246

–2–4

–6–8

x

y

(1,–8)

MATH 1314 - College Algebra - Review for Test 2 (Thomason) - p. 2 of 8

8. Determine the domain of

€

f (x) =2x − 3

x2 − 3x −10. Give your answer in set-builder notation.

9. For

€

3x2 − 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) (b) Subtract:

€

(−5 + 4i) − (3 − 2i)

(c) Multiply:

€

(3 − 2i)(−4 + i) (d) Divide:

€

4 + 3i5 − 2i

11. Solve

€

3x2 = 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) = −x2 − 2x + 3 is shown at the right. Solve

€

f (x) ≤ 0 and give your answer in interval notation.

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.

(–3,0) (1,0)

(0,3)(–1,4)

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:

€

7x2 − 9x > 0 16. Solve the following inequality and give your answer in interval notation:

€

x2 + 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 12 x( ) , and

(d)

€

y = 2 f (−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.

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) = x2 − 3 (b)

€

f (x) = (x − 3)2 (c)

€

f (x) = x2 + x 21. The table at the right is a complete representation of f. Is f an even function, an odd function, or neither? Show or explain how you determined.

x –5 –3 –1 0 1 3 5 f(x) 8 4 2 0 2 4 8

1 x

y

1

–3 –2 –1 1 2 3

1

2

3

–1

–2

–3

–4

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? (b) Estimate the x-intercepts, assuming they are integers. (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) = −2x3 + 5x2 +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 →∞ . 24. For

€

f ( x) = −x4 + 2x2 − 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

27. Let

€

f ( x) =

4 for x < −2x2 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

€

x3 − 4x2 − 20x − 3 by

€

x + 3. 29. Divide

€

6x3 − x2 + 4x − 7 by

€

3x − 2 . 30. Is

€

x + 2 a factor of

€

f (x) = x3 + 5x2 + 3x − 6? Tell how you determined your answer. 31. What is the remainder when

€

x3 − 3x2 + 4x − 5 is divided by

€

x − 2?

x

y

1

1

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:

€

7x3 − 5x2 − 21x +15 = 0 . Simplify your answers including removing perfect squares from under square roots and reducing fractions, when possible. 34. Solve exactly for x:

€

x2 = 4x −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:

€

x6 =10x3 − 21. Simplify your answers including removing perfect squares from under square roots and reducing fractions, when possible. Write any complex solutions in standard form.

36. Find the zeros of

€

f (x) = 7x3 + 5x2 +12x − 4 given that

€

27

is a zero.

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? (b) How many different imaginary zeros does

€

f (x) have?

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

y

3

3

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) = x3 − 3x2 + 9x − 27 . (b) Write

€

f (x) in completely factored form. 41. (a) Find all the zeros of

€

f ( x) = 4x3 + 32x . (b) Write

€

f (x) in completely factored form. 42. (a) Find all the zeros of

€

f ( x) = 3x4 − 6x3 +12x2 . (b) Write

€

f (x) in completely factored form. Section 4.7 (Polynomial Inequalities ) 43. Solve for x and give your solution in interval notation:

€

2x2 +15x < x3 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)

2. (a) (1,0), (5,0) (b) (0,10) (c) (3,–8) (d)

€

x = 2

€

x = 3

MATH 1314 - College Algebra - Review for Test 2 (Thomason) - p. 7 of 8

3.

€

f ( x) = −2( x +1)2 + 8 4.

€

f ( x) = 2x2 − 4x − 6 5. (a)

€

3123

ft (b)

€

4 + 193

sec

6. (a)

€

3 ± 5 (b)

€

3 ± 72

7.

€

4 ± 2 5 8.

€

x x ≠ −2,5{ } 9. (a) 1 (b) 2

10. (a)

€

−3 − i (b)

€

−8 + 6i (c)

€

−10 +11i

(d)

€

1429

+2329

i

11.

€

1±3

3i

12. (a) 2 (b) 0 (c) 1 13.

€

(−∞,−3][1,∞) 14. (a)

€

(−∞,−1] (b)

€

[−1,∞)

15.

€

(−∞,0)97

,∞⎛

⎝ ⎜

⎞

⎠ ⎟

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. (b) Neither, because

€

f (−x) = (−x − 3)2 = x2 + 6x + 9 but

€

f ( x) = ( x − 3)2 = x2 − 6x + 9 so

€

f (−x) ≠ f ( x) and

€

f (−x) ≠ − f (x).

(c) Neither, because

€

f (−x) = (−x)2 + (−x) = x2 − x but

€

f ( x) = x2 + 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

1 x

y

117(a)

1 x

y

117(c)

1 x

y

117(b)

1 x

y

117(d)

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.

27. (a) 4 (b) 9 28.

€

x2 − 7x +1−6

x + 3

(c) (d)

3

(d)

29.

€

2x2 + x + 2 −3

3x − 2

30. Yes, because the remainder given by

€

f (2) or division is 0. 31.

€

−1

32.

€

f ( x) =12

( x +1)2(x − 2)(x − 4)

33.

€

± 3 ,

€

57

34.

€

2 ± 3i 35.

€

33 ,

€

73 36.

€

27

, −12

±7

2i 37. (a) 2 (b) 2

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) = 3x2 x − 1− i 3( )⎡ ⎣ ⎢

⎤ ⎦ ⎥ x − 1+ i 3( )⎡ ⎣ ⎢

⎤ ⎦ ⎥

43.

€

(−3,0)(5,∞) 44.

€

[−4,2]{5}