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Resources

Transformations of quadratic functions

Use the remainder theorem to find

P(3)

P(x)=2x^{3}-4x^{2}-2x+9

. Specifically, give the quotient and the remainder for the associated division and the value of

P(3)

\square

=\square \square

P(3)=

\square

Given the function

f

f(-1), f(0), f(2)

f(4)

\newline

f(x)=\left\{\begin{array}{cc} x^{2}-3 & \text { if } x<2 \\ 4+|x-6| & \text { if } x \geq 2 \end{array}\right.

\newline

\begin{aligned} f(-1) & =\ \\ f(0) & =\ \\ f(2) & =\ \\ f(4) & =\\ \end{aligned}

Rewrite the function

f(x)=3(x-4)^{2}+2

f(x)=a x^{2}+b x+c

\newline

f(x)=

The product of two numbers is the greatest

2

-digit number. If one of the numbers is

-11

, find the other number.

f(x)= \frac{4x-k}{2}

g(x)=4x-1

f(g(x))=8x

, what is the value of

k

\newline

-4

\newline

0

\newline

2

\newline

4

14y^{3}

(7y^{2})(2y)

. Bilal factored

14y^{3}

(10y)(4y^{2})

. Which of them factored

14y^{3}

\newline

1

\newline

\newline

\newline

(C) Both Hiro and Bilal

\newline

(D) Neither Hiro nor Bilal

Joint equation of two lines, through

(2, -3)

, perpendicular to two lines

3x^{2}+xy-2y^{2}=0

A=\{1,2,3,4\},B=\{3,4,5,6\}, U=\{1,2,3,4,5,6\}

then prove De Morgan's law.

\newline

De Morgan's law: first Law:

(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}

(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}

a,b,c,k

be rational numbers such that

k

is not a perfect cube.

\newline

a+bk^{\frac{1}{3}}+ck^{\frac{2}{3}}

then prove that

a=b=c=0

g(x)=x^{5}+3 x

h

be the inverse function of

g

g(1)=4

\newline

h^{\prime}(4)=

h(x)=\frac{1}{4} x^{3}+2 x-1

g

be the inverse function of

h

h(2)=5

\newline

g^{\prime}(5)=

h(x)=7-x-2 x^{5}

f

be the inverse function of

h

h(-1)=10

\newline

f^{\prime}(10)=

f(x)=x^{3}+2 x-1

g

be the inverse function of

f

f(2)=11

\newline

g^{\prime}(11)=

h(x)=5-x-x^{3}

f

be the inverse function of

h

h(-1)=7

\newline

f^{\prime}(7)=

f(x)=\frac{1}{2} x^{3}+3 x-4

h

be the inverse function of

f

f(-2)=-14

\newline

h^{\prime}(-14)=

Chun was asked whether the following equation is an identity:

\newline

3(2 x-1)^{2}+27=36 x^{2}-36 x+36

\newline

He performed the following steps:

\newline

3(2 x-1)^{2}+27

\newline

\stackrel{\text { Step } 1}{\hookrightarrow}=3(2 x-1)(2 x-1)+27

\newline

\stackrel{\text { Step } 2}{\hookrightarrow}=(6 x-3)(6 x-3)+27

\newline

\stackrel{\text { Step } 3}{\hookrightarrow}=36 x^{2}-18 x-18 x+9+27

\newline

\stackrel{\text { Step } 4}{\hookrightarrow}=36 x^{2}-36 x+36

\newline

For this reason, Chun stated that the equation is a true identity.

\newline

Is Chun correct? If not, in which step did he make a mistake?

\newline

1

\newline

(A) Chun is correct.

\newline

(B) Chun is incorrect. He made a mistake in step

2

\newline

(C) Chun is incorrect. He made a mistake in step

3

\newline

(D) Chun is incorrect. He made a mistake in step

4

Mustafa was asked whether the following equation is an identity:

\newline

x(9 x+3)+3(x+3)=9(x+1)^{2}

\newline

He performed the following steps:

\newline

x(9 x+3)+3(x+3)

\newline

\stackrel{\text { Step } 1}{\hookrightarrow}=9 x^{2}+6 x+9

\newline

\stackrel{\text { Step } 2}{\hookrightarrow}=(3 x+3)^{2}

\newline

\stackrel{\text { Step } 3}{\hookrightarrow}=(3(x+1))^{2}

\newline

\stackrel{\text { Step } 4}{\hookrightarrow}=9(x+1)^{2}

\newline

For this reason, Mustafa stated that the equation is a true identity.

\newline

Is Mustafa correct? If not, in which step did he make a mistake?

\newline

1

\newline

(A) Mustafa is correct.

\newline

(B) Mustafa is incorrect. He made a mistake in step

1

\newline

(C) Mustafa is incorrect. He made a mistake in step

2

\newline

(D) Mustafa is incorrect. He made a mistake in step

4

Mabel was asked whether the following equation is an identity:

\newline

(4 x-3)(x-2)^{2}=4 x^{3}-19 x^{2}+4 x-12

\newline

She performed the following steps:

\newline

(4 x-3)(x-2)^{2}

\newline

\stackrel{\text { Step } 1}{\hookrightarrow}=(4 x-3)(x-2)(x-2)

\newline

\stackrel{\text { Step } 2}{\hookrightarrow}=(4 x-3)\left(x^{2}-2 x-2 x+4\right)

\newline

\stackrel{\text { Step } 3}{\hookrightarrow}=(4 x-3)\left(x^{2}-4 x+4\right)

\newline

\stackrel{\text { Step } 4}{\hookrightarrow}=4 x^{3}-16 x^{2}+16 x-3 x^{2}-12 x-12

\newline

\stackrel{\text { Step } 5}{\hookrightarrow}=4 x^{3}-19 x^{2}+4 x-12

\newline

For this reason, Mabel stated that the equation is a true identity.

\newline

Is Mabel correct? If not, in which step did she make a mistake?

\newline

1

\newline

(A) Mabel is correct.

\newline

(B) Mabel is incorrect. She made a mistake in step

2

\newline

(C) Mabel is incorrect. She made a mistake in step

3

\newline

(D) Mabel is incorrect. She made a mistake in step

4

4 x-x^{2} y+y^{3}=10

\newline

Find the value of

\frac{d y}{d x}

(1,2)

\newline

1

\newline

0

\newline

\frac{2}{3}

\newline

-1

\newline

-4

For the rotation

670^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

For the rotation

417^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

For the rotation

965^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

For the rotation

804^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

For the rotation

1366^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

For the rotation

-337^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

For the rotation

645^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

For the rotation

1025^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

For the rotation

966^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

For the rotation

440^{\circ}

, find the coterminal angle from

0^{\circ} \leq \theta<360^{\circ}

, the quadrant, and the reference angle.

\newline

The coterminal angle is

\square^{\circ}

, which lies in Quadrant

\square

, with a reference angle of

\square^{\circ}

3 x=5+9 x

F_{1}

has a land in the shape of a triangle with vertices at

P(0,0)

Q(1,1)

R(2,0)

. From this land, a neighbouring farmer

F_{2}

takes away the region which lies between the side

PQ

and a curve of the form

y=x^{n}

(n > 1)

. If the area of the region taken away by the farmer

F_{2}

30\%

\Delta

PQR, then the value of

n

The angles of a triangle are

x, y

40^{\circ}

. The difference between the two angles

x

y

30^{\circ}

x

y

y=x^{2}

is shifted up by

2

units and to the right by

3

\newline

What is the equation of the new parabola?

\newline

y=\square

\frac{8 x^{2}-6 x+3}{(2 x-1)^{2}}=A+\frac{B}{2 x-1}+\frac{C}{(2 x-1)^{2}}

, then A= _____, B= _____, C= _____

a, b, c

be the three rational numbers where

a=\frac{2}{3} \quad b=\frac{4}{5}

c=\frac{-5}{6}

\newline

\mathrm{a}+(\mathrm{b}+\mathrm{c})=(\mathrm{a}+\mathrm{b})+\mathrm{c}

(Associative property of addition)

(y^{2}-t^{2})(y+k)

can be written as

y^{3}+36 y^{2}-9 y+s

t, k

s

are constants. What is the value of

s

\newline

1

\newline

-324

\newline

-54

\newline

54

\newline

324

a x-a(a+b)=-x-(1+a b)

\newline

Use the quotient rule to evaluate

h^{\prime}(a)

for the given function

h(x)

a

\newline

h(x)=\frac{3 x^{2}}{x^{2}+9 x+9} \quad a=-3

1+2i

k(x)=x^{4}-6x^{3}+26x^{2}-46x+65

, find the remaining zeroes

\newline

\begin{align*} 1+2i &= 0 & 1-2i &= 0 \ (x-(1+2i))(x-(7-2i)) & \end{align*}

Choose all of the functions that are nonlinear.

\newline

y=4-2x

\newline

y=\frac{5}{at}-2

\newline

y=\frac{21}{31}x-7

\newline

y=8(x-1)

\newline

y=x(2x+4)

The quadratic equation

x^{2}-8x=-20

is to be solved by completing the square. Which equation would be a step in that solution?

\newline

(x-4)^{2}=4

\newline

x^{2}-8x+16=-20+16

\newline

x^{2}-8x+20=0

\newline

x^2-8x+16=-20

f

f(x)=\frac{7}{2x^{2}-4x}

\newline

f(x+6)

\newline

Write your answer without parentheses, and simplify it as much as possible.

\newline

f(x+6)=

\square

Solve the equation by factoring:

\newline

2 x^{3}-18 x^{2}+16 x=0

\newline

x=

T(t)=25+65 \cdot(0.78)^{t}

\newline

A pot of soup is heated and then left to cool in a room with a constant temperature. The equation gives the temperature of the soup,

T(t)

, in degrees Celsius

\left({ }^{\circ} \mathrm{C}\right), t

minutes after it is heated. What is the initial temperature of the soup before it begins to cool?

\newline

1

\newline

25^{\circ} \mathrm{C}

\newline

65^{\circ} \mathrm{C}

\newline

78^{\circ} \mathrm{C}

\newline

90^{\circ} \mathrm{C}

Solve the equation

4 x^{2}-10 x+8=-x^{2}+22

to the nearest tenth.

\newline

x=

Evaluate the left hand side to find the value of

a

in the equation in simplest form.

\newline

x^{3} x^{\frac{1}{5}}=x^{a}

\newline

Evaluate the left hand side to find the value of

a

in the equation in simplest form.

\newline

x^{2} x^{\frac{3}{2}}=x^{a}

\newline

Evaluate the left hand side to find the value of

a

in the equation in simplest form.

\newline

x^{3} x^{\frac{5}{2}}=x^{a}

\newline

Evaluate the left hand side to find the value of

a

in the equation in simplest form.

\newline

x^{2} x^{\frac{2}{3}}=x^{a}

\newline

Factor completely:

\newline

(3 x+5)^{2}(2 x+7)+(3 x-7)(3 x+5)

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