Free Printable Alg II Exam Review Sem 1

1.

For $y=x^2-5$ , what are the coordinates of the vertex?

a)

(0, −5)

b)

(2, 0)

c)

(−2, 0)

d)

(0, 2)

2.

For $y=-3x^2+4$ , on which intervals is the function increasing?

a)

(−∞, 0) and (0, ∞)

b)

(−∞, 0) only

c)

(−∞, ∞) excluding 0

d)

(0, ∞) only

3.

For $y=\left|x+3\right|+1$

what are the coordinates of the vertex?

a)

(−3, 1)

b)

(3, 1)

c)

(−3, −1)

d)

(3, −1)

4.

For y = |x − 4| − 2, on which interval is the graph decreasing?

a)

x > 4 only

b)

x < 4 only

c)

x ≠ 4 only

d)

x ≥ 4 and x ≤ 4

5.

Which of the following graphs is the graph of

$f\left(x\right)=$

$x+3,\ \ \ x<-1$
$\frac{1}{4}x^2+3,\ \ \ x\ge-1$

a)

b)

c)

d)

6.

Which of the piecewise functions matches this graph?

a)

b)

c)

d)

7.

What is the domain of the graph?

a)

1≤ x ≤ 4

b)

1 < x < 4

c)

1≤ y ≤ 4

d)

1 < y < 4

8.

Maria’s monthly expenses follow an arithmetic sequence: April $1500, May $1200, June $900, July $600. What is the explicit formula $a_n=a_1+d\left(n-1\right)$ for the expenses in month n, taking April as n = 1?

a)

$a_n$ =1500− 300(n − 1)

b)

$a_n$ = 1500 − 400(n − 1)

c)

$a_n$ = 1500 − 200(n − 1)

d)

$a_n$ = 1500 − 100(n − 1)

9.

Use the explicit formula to find the first four terms of the sequence.
$a_n=3+5\left(n-1\right)$

a)

3, 15, 75, 375

b)

3, 8, 13, 18

c)

5, 8, 11, 14

d)

5, 15, 45, 135

10.

The sequence 28, 34, 40, 46, 52, ..., 88 has 11 terms. Find the sum of the 11 terms.

$S_n=\frac{n\left(a_1+a_n\right)}{2}$

a)

550

b)

638

c)

610

d)

288

11.

Mya flies her drone 15 m, then 30 m, then 45 m, continuing the pattern. If the last test flight was 90 m, what is the total distance flown across all tests? Use $S_n=\frac{n\left(a_1+a_n\right)}{2}$

a)

315 meters total

b)

525 meters total

c)

630 meters total

d)

450 meters total

12.

Solve the equation $|x - 2|$ + 1 = $(x - 1)^2$ . How many real solutions does it have?

a)

No real solutions

b)

Exactly three real solutions

c)

Exactly two real solutions

d)

Exactly one real solution

13.

Find the solution(s) to the system by graphing.

$y=x^2-3$

$y=x-3$

a)

(0,-3)(1,-2)

b)

(0,-3)

c)

(-2,0)(2,0)

d)

No solution

14.

Find the x-coordinate(s) to the linear-quadratic system below: $x^2+y=6$

$y=5x$

a)

1 and -1

b)

2 and -2

c)

1 and -6

d)

0 and 0

15.

Solve the system

3x + y = 12

x − y = 4

What is the value of x?

a)

x = 2

b)

x = 4

c)

x = 8

d)

x = 6

16.

Solve the system of equations.

x-y=11

2x+y=19

a)

(10,-1)

b)

(-1,10)

c)

(-1,-10)

d)

(-10,-1)

17.

There are 15 animals in a barn. These animals are horses and chickens. There are 44 legs in all. Which system of equations represents the situation?

a)

x + y = 15
4x + 2y = 44

b)

4x + 2y = 15
x + y = 44

c)

x = 2y + 44
4x = y + 15

d)

2x - 4y = 44
x - y = 15

18.

You and a friend go to Chick-Fil-A. You order 2 chicken sandwiches and 5 cookies. Your bill is $17. Your friend orders 3 chicken sandwiches and 2 cookies. Their bill is $20. How much does a chicken sandwich and a cookie cost?

a)

A sandwich costs $6 and a cookie costs $1.

b)

A sandwich costs $1 and a cookie costs $6.

c)

A sandwich costs $5 and a cookie costs $2.

d)

A sandwich costs $2 and a cookie costs $5.

19.

For y = $x^2 + 4x + 5$ , what is the axis of symmetry?

a)

x = −2

b)

x = −4

c)

x = 2

d)

x = 4

20.

For y = $x^2 + 4x + 3$ , identify the vertex.

a)

(-2, -1)

b)

(2, 7)

c)

(-4, 3)

d)

(0, 3)

21.

For y = $x^2 + 3x + 2$ , what is the y-intercept?

a)

(0, 2)

b)

(0, 3)

c)

(0, 1)

d)

(0, 0)

22.

For y = $x^2 + 4x + 3$ , determine the minimum or maximum value.

a)

Maximum 3 at x = 0

b)

Maximum 4 at x = 1

c)

Minimum -1 at x = -2

d)

Minimum 3 at x = 0

23.

Describe the transformation from $y = x^2$ to $y = 3(x - 4)^2 + 2$ .

a)

Right 4, up 2, vertical stretch by 3

b)

Left 4, down 2, vertical stretch by 3

c)

Right 4, up 2, vertical shrink by 3

d)

Left 4, up 2, vertical stretch by 3

24.

Describe the transformation:

f\left(x\right)=\left(x+3\right)^2-2

a)

right 3, down 2

b)

up 3, left 2

c)

left 3, up 2

d)

left 3, down 2

25.

Which equation would shift the quadratic function

f\left(x\right)=x^2

up 3 units and to the right 4 units.

a)

$f\left(x\right)=\left(x+3\right)^2+4$

b)

$f\left(x\right)=3\left(x-4\right)^2$

c)

$f\left(x\right)=\left(x-4\right)^2+3$

d)

$f\left(x\right)=\left(x+4\right)^2+3$

26.

Write the vertex form of a parabola with vertex (1, 3) passing through (2, −6).

$Formula:f(x)=a(x-h)^2+k$

a)

$y = −3(x − 1)^2 + 9$

b)

$y = −9(x + 1)^2 + 3$

c)

$y = −9(x − 1)^2 + 3$

d)

$y = 9(x − 1)^2 − 3$

27.

A parabola has a vertex at (6, 5) and passes through (10, -11). Which equation represents this parabola in vertex form?

Formula: $f\left(x\right)=a\left(x-h\right)^2+k$

a)

-11 = a(10 - 6)² + 5

b)

y = -(10 - h)² + k

c)

y = -(x - 6)² + 5

d)

y = (x - 6)² + 5

28.

Find the standard form of the quadratic passing through (1, 5), (3, 7), and (6, 25).

a)

$y = x^2 - 4x + 9$

b)

$y = x^2 - 2x + 6$

c)

$y = x^2 - 3x + 7$

d)

$y = x^2 - x + 5$

29.

Find the standard form of the quadratic passing through (2, 8), (4, 12) and (5, 23).

a)

$y=3x^2-16x+28$

b)

$y=4.5x-2.4$

c)

$y=3x^2-16x-28$

d)

$y=x^2-2x+8$

30.

Solve the quadratic equation $x^2-7x+12=0$ . Which statement gives the solutions?

Quadratic Formula: $x=\frac{-b\pm\sqrt[]{b^2+-4ac}}{2a}$

a)

x = 3 and x = 4

b)

x = 2 and x = 6

c)

x = 1 and x = 12

d)

x = −3 and x = −4

31.

Solve $x^2 − 12x + 27 = 0$ . What are the roots?

Quadratic Formula: $x=\frac{-b\pm\sqrt[]{b^2+-4ac}}{2a}$

a)

x = 6 ± √9 over 1

b)

x = 3 and x = 9

c)

x = 5 and x = 7

d)

x = 6 ± √9

32.

Solve $3x^2 - 6x + 12 = 0$ . What is the solution ?

Quadratic Formula: $x=\frac{-b\pm\sqrt[]{b^2+-4ac}}{2a}$

a)

x = 1 ± √3

b)

x = 1 ± √6

c)

x = 1 ± i√6

d)

x = 1 ± i√3

33.

Solve $x^2 + 6x + 9 = 0$ . Which description is correct?

a)

No real solutions; complex only

b)

Two complex roots, x = -3 ± 2i

c)

A repeated real root at x = -3

d)

Two distinct real roots, x = 0 and x = -9

34.

Simplify the complex sum (5 + 3i) + (2 − 7i). Choose the result in a + bi form.

a)

7 − 4i

b)

3 − 10i

c)

7 + 10i

d)

3 + 4i

35.

Simplify the complex sum (5 + 2i) + (−3 + 7i). What is the result?

a)

−8 + 9i

b)

8 + 5i

c)

2 + 9i

d)

2 + 5i

36.

Simplify the complex difference (6 − 7i) − (6 − 4i). What is the result?

a)

0 − 3i

b)

0 + 3i

c)

6 − 11i

d)

12 − 3i

37.

Compute 3i(12 + 7i). What is the product?

a)

36i -21

b)

21 + 36i

c)

36 + 21i

d)

$36i + 21i^2$

38.

Multiply:
(4 – 3i)(-7 – 2i)

a)

A. -23 + 13i

b)

B. -23 - 29i

c)

C. -34 + 13i

d)

D. -34 - 29i

39.

Rewrite f(x) = $3x^3-8x+2x^2-x^5+2x^4$ in standard form. Then identify its degree and leading coefficient.

a)

Standard form $−x^5+3x^3+2x^2+2x^4−8x$ ; degree 5; LC 2

b)

Standard form $2x^4-x^5+3x^3+2x^2-8x$ ; degree 5; LC 2

c)

Standard form $3x^3+2x^2-8x-x^5+2x^4$ ; degree 4; LC 2

d)

Standard form $−x^5+2x^4+3x^3+2x^2−8x$ ; degree 5; LC −1

40.

For a polynomial with odd degree and negative leading coefficient, which end behavior is correct?

a)

As x → −∞, y → ∞;

as x → ∞, y → ∞

b)

As x → −∞, y → 0

as x → ∞, y → 0

c)

As x → −∞, y → −∞

as x → ∞, y → ∞

d)

As x → −∞, y → ∞

as x → ∞, y → −∞

41.

Consider h(x) = $x^3 + 3x^2$ . Which statement about its relative extrema is true?

a)

A relative minimum at x = 0 and a relative maximum at x = 2

b)

A relative maximum at x = 0 and a relative minimum at x = −2

c)

A relative minimum at x = −2 and a relative maximum at x = 0

d)

A relative maximum at x = −2 and a relative minimum at x = 0

42.

Find the zeros of h(x) = $x^3 + 3x^2$ .

a)

x = 3, x = −1

b)

x = 0 only

c)

x = 0, x = −3

d)

x = −3 only

43.

Simplify $(2x^3 + 3x^2 + 4) − (6x^3 − x^2 − 5x)$ .

a)

$4x^3 + 2x^2 - 5x + 4$

b)

$-4x^3 + 4x^2 + 5x + 4$

c)

$−4x^3 + 2x^2 − 5x + 4$

d)

$4x^3 - 4x^2 + 5x - 4$

44.

Expand and simplify (4x + y)(3x − 2y + 5).

a)

$12x^2-5xy-20x-2y^2+5y$

b)

$12x^2-5xy+20x+2y^2+5y$

c)

$12x^2+5xy+20x+2y^2-5y$

d)

$12x^2-5xy+20x-2y^2+5y$

45.

Expand $(3x + 2y)^5$ using the Binomial Theorem. Choose the correctly expanded polynomial.

a)

$243x^5 + 810x^4y + 1080x^3y^2 + 960x^2y^3 + 400xy^4 + 64y^5$

b)

$243x^5 + 405x^4y + 360x^3y^2 + 160x^2y^3 + 40xy^4 + 4y^5$

c)

$243x^5 + 810x^4y + 1080x^3y^2 + 720x^2y^3 + 240xy^4 + 32y^5$

d)

$243x^5 + 540x^4y + 720x^3y^2 + 480x^2y^3 + 160xy^4 + 16y^5$

46.

Factor completely: $4x^2 - 144$ .

a)

$(2x - 12)^2$

b)

(x − 12)(x + 12)

c)

$4(x − 6)^2$

d)

4(x + 6)(x − 6)

47.

Factor completely: $216 + 343y^3$ .

a)

$(6 − 7y)(36 + 42y + 49y^2)$

b)

$(6 + 7y)(36 + 42y + 49y^2)$

c)

$(6 + 7y)(36 - 42y + 49y^2)$

d)

$(6 − 7y)(36 − 42y + 49y^2)$

48.

Perform polynomial long division and simplify: $(x^4 - 25x^2 + 144) ÷ (x - 4)$ . Choose the correct quotient and remainder.

a)

$x^3 + 4x^2 - 9x - 36$ , remainder -0·1

b)

$x^3 + 4x^2 - 9x - 36$ , remainder -0x

c)

$x^3 + 4x^2 - 9x - 36$ , remainder -0

d)

$x^3 + 4x^2 - 9x - 36$ , remainder 0

49.

Given f(x) = $x^3 − 8x^2 + 16x$ , determine all real zeros. Select the correct set.

a)

x = −2, x = 0, x = 8

b)

x = 0, x = 2, x = 8

c)

x = −4, x = 0, x = 4

d)

x = 0, x = 4, x = 4

50.

Use Pascal’s Triangle to find the coefficient of $x^3y^2$ in $(3x + 2y)^5$ .

a)

720

b)

1080

c)

405

d)

540

51.

Solve for x: $−5x^3 + 4x − 12 = −6x^3 + 3x^2.$

a)

$x=\pm2\iota\ or\ x=−3$

b)

$x=\pm2\iota\ or\ x=0$

c)

$x=0\ or\ x=\pm3\iota$

d)

$x=\pm2\iota\ or\ x=3$

52.

Which inequality solution set best matches $0 > 4x^3 + 8x^2 - x - 2$ ?

a)

$x<−1\ and\ \frac{−1}{2}<x<0$

b)

$x<−2\ and\ \frac{1}{2}<x<1$

c)

x < −1 and 0 < x < 2

d)

x < −2 and -1/2< x < 1/2

53.

Using the Rational Root Theorem, what are all possible rational roots of $4x^3 + 8x^2 - x - 2 = 0$ ?

a)

±1, ±2, ± $\frac{1}{2}$

b)

±1, ±2, ± $\frac{1}{2}$ , ± $\frac{1}{3}$

c)

±1, ±2, ±4, ± $\frac{1}{2}$

d)

±1, ±2, ± $\frac{1}{2}$ , ± $\frac{1}{4}$

54.

Which set lists P and Q correctly for $4x^3 + 8x^2 - x - 2 = 0$ when applying p/q ?

a)

P: factors of 4; Q: factors of −1

b)

P: multiples of −2; Q: primes of 4

c)

P: factors of 4; Q: factors of −2

d)

P: factors of −2; Q: factors of 4

55.

Which x-value is an actual rational root of $4x^3 + 8x^2 - x - 2 = 0$ ?

a)

x = 0

b)

x = 2

c)

x = $-\frac{1}{2}$

d)

x = 1

56.

Write a polynomial whose roots include 1 + 6i and 2.

a)

$x^3 - 3x^2 + 37x - 40$

b)

$x^3-4x^2+41x-74$

c)

$x^3 - 4x^2 + 37x - 40$

d)

$x^3 - 4x^2 + 40x - 52$

57.

If 1 + 6i is a root of a polynomial with real coefficients, which statement is necessarily true?

a)

2 − 6i is also a root

b)

1 − 6i is also a root

c)

6 + i is also a root

d)

−1 − 6i is also a root

58.

For f(x) = $\frac{3}{x-6}−4$ , what is the vertical asymptote?

a)

x = 0

b)

x = 6

c)

x = −4

d)

x = 3

59.

For f(x) = $\frac{3}{x-6}-4$ , what is the horizontal asymptote?

a)

y = 0

b)

y = 6

c)

y = 3

d)

y = −4

60.

For f(x) = $\frac{3}{x-6}-4$ , which is the domain?

a)

All real x except x ≠ 6

b)

All real x except x ≠ 3

c)

All real x except x ≠ −4

d)

All real x except x ≠ 0

61.

For f(x) = $\frac{3}{x-6}-4$ , which is the range?

a)

All real y except y ≠ −4

b)

All real y except y ≠ 0

c)

All real y except y ≠ 6

d)

All real y except y ≠ 3

62.

Which transformation maps y = $\frac{1}{x}$ to f(x) = $\frac{3}{x-6}−4$ ?

a)

Right 6, up 4, vertical shrink by 3

b)

Left 6, up 4, vertical stretch by 3

c)

Left 6, down 4, vertical shrink by 3

d)

Right 6, down 4, vertical stretch by 3

63.

Given g(x) = $\frac{3}{x^2−2x−8}$ , what are the vertical asymptotes?

a)

x = 2 and x = −4

b)

x = −2 and x = 4

c)

x = 6 and x = −4

d)

x = 0 and x = 4

64.

For g(x) = $\frac{3}{x^2−2x−8}$ , what is the horizontal asymptote?

a)

y = −3

b)

y = 3

c)

y = 1

d)

y = 0

65.

$xy=k\ \ \ \ \ y=\frac{k}{x}$

If y varies inversely with x and y = −4 when x = 6, what is y when x = 10?

a)

y = 2.4

b)

y = −2.4

c)

y = 6.7

d)

y = −6.7

66.

Simplify the rational expression: $\frac{x^2−16x+64}{x^2−64}$ ÷ $\frac{x^2−6x−16}{x+2}$ . Assume all denominators are nonzero.

a)

$\frac{x-4}{x+4}$

b)

$\frac{1}{x+8}$

c)

$\frac{\left(x-8\right)\left(x-8\right)}{x+8}$

d)

$\frac{(x−8)(x−4)}{(x−8)(x−2)}$

67.

Multiply and simplify: $\frac{x^2 + 8x + 15}{x - 6}$ · $\frac{x^2 - 9x + 18}{x^2 - 9}$ . Factor completely before canceling.

a)

$\frac{(x+3)(x+5)}{x−3}$

b)

$\frac{(x+3)(x+5)}{x−6}$

c)

$x+5$

d)

$\ \frac{x+5}{x-6}$

68.

Divide and simplify: $\frac{9x^2y^2}{5z^4}\div\frac{12x^4y^2}{50xy^2z^2}$ . Assume variables represent nonzero values.

a)

$\frac{15 z^2}{4 x^3}$

b)

$\frac{5 z^2}{4 x^3}$

c)

$\frac{15y^2}{2xz^2}$

d)

$\frac{5}{4 x^3 z^2}$

69.

Find the least common denominator (LCD) of the expressions $\frac{2}{x^2y}$ and $\frac{5}{x}$ .

a)

$x^2 y$

b)

$x^3y$

c)

$x y^2$

d)

$x^2 y^2$

70.

Add the rational expressions $\frac{2}{x^2 y} + \frac{5}{x}$ . Assume x ≠ 0 and y ≠ 0.

a)

$\frac{2+5x^2}{x^2y}$

b)

$\frac{2+5y}{x^2y}$

c)

$\frac{2+5x^2y}{x^2y}$

d)

$\frac{2+5xy}{x^2y}$

71.

For $\frac{3}{x^2+x−12}$ − $\frac{2}{x^2+6x+8}$ , first factor each denominator to prepare for combining. Which factorizations are correct?

a)

(x+3)(x+4) and (x+2)(x+6)

b)

(x+6)(x−2) and (x+6)(x+8)

c)

(x+4)(x−3) and (x+4)(x+2)

d)

(x−3)(x−4) and (x+2)(x+4)

72.

Compute $\frac{3}{x^2+x−12}$ − $\frac{2}{x^2+6x+8}$ after factoring and using the LCD. Simplify your final numerator.

a)

$\frac{2x−1}{(x−3)(x+4)\left(x+2\right)}$

b)

$\frac{x−5}{(x−3)(x+4)\left(x+2\right)}$

c)

$\frac{x+12}{(x−3)(x+4)\left(x+2\right)}$

d)

$\frac{x}{(x−3)(x+4)\left(x+2\right)}$

73.

Solve the rational equation $\frac{9}{x−7}$ − $\frac{7}{x−6}$ = $\frac{13}{x^2−13x+42}$ . State x that satisfies all domain restrictions.

a)

x = 13

b)

x = 9

c)

x = 7

d)

x = 1

74.

When solving $\ \frac{9}{x-7}-\frac{7}{x-6}=\frac{13}{x^2-13x+42}$ , what is the least common denominator?

a)

(x−7)(x−6)

b)

(x−7)(x−6)(x−13)

c)

$(x−7)(x−6)(x^2−13x+42)$

d)

(x−6)(x−13)

75.

Simplify the complex fraction $\frac{x^2−9}{3x−6}$ ÷ $\frac{x^2+5x+6}{x^2−x−2}$ . Assume x values avoid zero denominators.

a)

$\frac{x+3}{3\left(x+2\right)}$

b)

$\frac{x−3}{3(x+2)}$

c)

$\frac{x+3}{3(x−2)}$

d)

$\frac{\left(x−3\right)\left(x+1\right)}{3(x+2)}$

76.

Rewrite $81^{\frac{1}{2}}$ as a radical expression.

a)

$\sqrt{81}$

b)

$\sqrt[3]{81}$

c)

$2\sqrt{81}$

d)

$81^2$

77.

Simplify $\sqrt{72}$ .

a)

$6\sqrt{2}$

b)

$8\sqrt{3}$

c)

$12$

d)

$2\sqrt{18}$

78.

Which of the following is equivalent to $x^{\frac{5}{3}}$ ?

a)

$\sqrt[3]{x^5}$

b)

$(\sqrt[3]{x})^5$

c)

$x^5 \cdot x^3$

d)

Both $\sqrt[3]{x^5}$ $\&\ (\sqrt[3]{x})^5$

79.

What is the domain of the function $f(x) = \sqrt{5-x}$ ?

a)

$x \leq 5$

b)

$x < 5$

c)

$x \geq 5$

d)

All real numbers

80.

Which of the following describes the transformation from $f(x) = \sqrt{x}$ to $h(x) = \sqrt{x+3} - 5$ ?

a)

Left 3 units, down 5 units

b)

Right 3 units, up 5 units

c)

Left 5 units, down 3 units

d)

Right 5 units, up 3 units

81.

Solve for $x$ : $\sqrt{x+3} = 5$ .

a)

$x = 22$

b)

$x = 25$

c)

$x = 20$

d)

$x = 18$

82.

Solve for $y$ : $y^{\frac{1}{2}} = 7$ .

a)

$y = 14$

b)

$y = 49$

c)

$y = 7$

d)

$y = 3.5$

83.

Which of the following is a solution to the equation $\sqrt{x+5} = x - 1$ ?

a)

$x = 2$

b)

$x = 3$

c)

$x = 4$

d)

$x = 5$

84.

If $f(x) = x^2$ and $g(x) = \sqrt{x}$ , what is $(g \circ f)(3)$ ?

a)

$3$

b)

$6$

c)

$9$

d)

$\sqrt{9}$

85.

If $f(x) = 3x - 2$ and $g(x) = x + 4$ , find $(f + g)(2)$ .

a)

$8$

b)

$10$

c)

$6$

d)

$12$

86.

Solve for $x$ : $\sqrt{x+3}-1=x$ .

a)

$x = 2$

b)

$x = -3$

c)

$x = 1$

d)

$x = 4$

87.

If $h(x) = 2x + 5$ , what is $h(3)$ ?

a)

$6$

b)

$7$

c)

$8$

d)

$11$

88.

If $g(x) = x^2 + 5$ , what is $g^{-1}(x)$ ?

a)

$g^{-1}(x) = x - 5$

b)

$g^{-1}(x) = x^2 - 5$

c)

$g^{-1}(x) = \sqrt{x-5}$

d)

$g^{-1}(x) = \sqrt{x}+5$

89.

What is the range of the function $f(x) = \sqrt{x-2}$ ?

a)

$y \leq 0$

b)

$y > 2$

c)

All real numbers

d)

$y \geq 0$

90.

Which of the following is the inverse of $f(x) = \sqrt{x-2}$ ?

a)

$f^{-1}(x) = x^2 + 2$

b)

$f^{-1}(x) = x^2 - 2$

c)

$f^{-1}(x) = \sqrt{x} + 2$

d)

$f^{-1}(x) = \sqrt{x-2}$

Alg II Exam Review Sem 1

Total questions: 90

Worksheet time: 2hrs 35mins

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