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Resources

Transformations of quadratic functions

A chemical is diluted out of a tank by pumping pure water into the tank and pumping the existing solution out of it, so the volume at any time

t

20+2t

z

of chemical in the tank decreases at a rate proportional to

z

and inversely proportional to the volume of solution in the tank. Which equation describes this relationship? Choose

1

\newline

\frac{dz}{dt}=-\frac{kz}{20+2t}

\newline

\frac{dz}{dt}=kz-\frac{1}{20+2t}

\newline

\frac{dz}{dt}=k(20+2t)-\frac{1}{z}

\newline

\frac{dz}{dt}=-\frac{k(20+2t)}{z}

A national health survey weighed a sample of

546

6-11

74

of them were overweight. They weighed a sample of

455

6-11

75

of them were overweight. Can you conclude that the proportion of boys who are overweight is less than the proportion of girls who are overweight?

\newline

p_{1}

denote the proportion of boys who are overweight and let

p_{2}

denote the proportion of girls who are overweight.

\newline

\alpha=0.10

level of significance and the

p_{\text{-value }}

f

be a function from

\mathbb{R}

\mathbb{R}

m

n

real numbers such that

mf(x-1) + nf(-x) = 2|x|+1

for all real numbers. If

f(-2) = 5

f(1) = 1

, then find the numbers

m

n

\log \frac{a}{b-c} = \log \frac{b}{c-a} = \log \frac{c}{a-b}

a^{a}\times b^{b}\times c^{c}=1

Write the following expression in the form

a(x-k)^{2}+h

\newline

x^{2}-6 x+4

f

f(x) = 2x^3 + 3x^2 + cx + 8

c

\newline

In the xy-plane, the graph of

f

intersects the x-axis at the three points

\newline

(-4, 0)

(-1, 0)

(p, 0)

. What is the value of

c

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-18

\newline

-2

\newline

f(x) = 2x^3 + 3x^2 + cx + 8

0

\newline

f(x) = 2x^3 + 3x^2 + cx + 8

1

\newline

Consider the functions

g

h

g(x)=4^{x}

h(x)=16^{x+2}

x y

-plane, what is the

x

-coordinate of the point of intersection of the graphs of

g

h

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-4

\newline

-2

\newline

0

\newline

2

x(1400 x+6)+2(2x+4)

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The expression shown above can be expressed as

cx^{2}+kx+m

c

k

m

are constants. What is the value of

k

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The daily low temperature in Guangzhou, China, varies over time in a periodic way that can be modeled by a trigonometric function.

\newline

The period of change is exactly

1

year. The temperature peaks around July

26

78^{\circ} \mathrm{F}

, and has its minimum half a year later at

49^{\circ} \mathrm{F}

. Assuming a year is exactly

365

26

\frac{206}{365}

of a year after January

1

\newline

Find the formula of the trigonometric function that models the daily low temperature

T

t

years after January

1

2015

. Define the function using radians.

\newline

T(t)=\square

Maya was asked whether the following equation is an identity:

\newline

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

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She performed the following steps:

\newline

(2 x+3)(x+1)

\newline

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

\newline

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

\newline

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

\newline

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

\newline

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

\newline

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

\newline

1

\newline

(A) Maya is correct.

\newline

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

1

\newline

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

2

\newline

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

4

f^{\prime}(x)=[f(x)]^{2}

f(0)=1

f(6)=1 / n

for some integer

n

\newline

n

\newline

n=

f^{\prime}(x)=-f(x)

f(1)=2

f(3)=m e^{n}

for some integers

m

n

\newline

m

n

\newline

\begin{array}{l} m= \square \\ n= \square \end{array}

For the rotation

1036^{\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

-17^{\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

687^{\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

478^{\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

1140^{\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

708^{\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

1211^{\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}

A curve is such that

\frac{d y}{d x}=\frac{6}{(2 x-1)^{2}}

P(2,9)

is a point on the curve. The normal to the curve at

P

y

Q

x

R

. Find the coordinates of the midpoint of

Q R

When graphed in the

xy

-plane, the function

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

y

(0,k)

. What is the value of

k

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1

\newline

1

\newline

2

\newline

3

\newline

4

Given the definitions of

f(x)

g(x)

below, find the value of

(g \circ f)(-7)

\newline

\begin{array}{l} f(x)=2 x+15 \\ g(x)=2 x^{2}+5 x+4 \end{array}

\newline

Given the definitions of

f(x)

g(x)

below, find the value of

g(f(-3))

\newline

\begin{array}{l} f(x)=x^{2}+5 x+13 \\ g(x)=-5 x+4 \end{array}

\newline

Given the definitions of

f(x)

g(x)

below, find the value of

f(g(4))

\newline

\begin{array}{l} f(x)=x^{2}+4 x+11 \\ g(x)=3 x-11 \end{array}

\newline

Given the definitions of

f(x)

g(x)

below, find the value of

g(f(-4))

\newline

\begin{array}{l} f(x)=2 x+13 \\ g(x)=x^{2}-3 x+8 \end{array}

\newline

Given the definitions of

f(x)

g(x)

below, find the value of

(g \circ f)(-2)

\newline

\begin{array}{l} f(x)=-x-5 \\ g(x)=2 x^{2}+4 x-6 \end{array}

\newline

Given the definitions of

f(x)

g(x)

below, find the value of

g(f(2))

\newline

\begin{array}{l} f(x)=x^{2}-7 x+13 \\ g(x)=-2 x+8 \end{array}

\newline

Given the definitions of

f(x)

g(x)

below, find the value of

(g \circ f)(-2)

\newline

\begin{array}{l} f(x)=x^{2}-x-11 \\ g(x)=-2 x-4 \end{array}

\newline

Given the definitions of

f(x)

g(x)

below, find the value of

g(f(-8))

\newline

\begin{array}{l} f(x)=x+13 \\ g(x)=x^{2}+5 x-11 \end{array}

\newline

h(x)=-5(x-1)^{2}+7

\newline

A fountain sprays a constant stream of water from a nozzle. The function shows

h

, the height of the stream of water in feet above the water surface, in relation to

x

, the horizontal distance in feet from the nozzle. What is the best interpretation of the number

7

in the function?

\newline

1

\newline

(A) The stream of water begins at an initial height of

7

feet above the surface.

\newline

(B) The stream of water will reach a maximum height of

7

feet above the surface.

\newline

(C) The stream of water will touch the surface at a distance of

7

feet horizontally from the nozzle.

\newline

(D) The stream of water will reach a maximum height at a distance of

7

feet horizontally from the nozzle.

Write an explicit formula that represents the sequence defined by the following recursive formula:

\newline

a_{1}=50 \text { and } a_{n}=\frac{1}{5} a_{n-1}

\newline

a_{n}=

f(x)=\left\{\begin{array}{lll} -(x-1)^{2}+1 & \text { for } & 1 \leq x<4 \\ -1 & \text { for } & x=4 \\ 6 x-32 & \text { for } & x>4 \end{array}\right.

\newline

f(4)

\newline

\newline

Google Classroom

\newline

(x-h)^{2}+(y-k)^{2}=r^{2}

\newline

The formula relates the

x

y

-coordinates of a point on a circle with a center at

(h, k)

r

. Which of the following equations correctly gives

y

x, h, k

r

\newline

1

\newline

y=r-x+h+k

\newline

y=r-(x-h)^{2}+k

\newline

y= \pm \sqrt{r^{2}-(x-h)^{2}}+k

\newline

y

0

\newline

Show calculator

Write an explicit formula for

a_{n}

n^{\text {th }}

term of the sequence

72,-24,8, \ldots

\newline

a_{n}=

Write an explicit formula for

a_{n}

n^{\text {th }}

term of the sequence

37,31,25, \ldots

\newline

a_{n}=

Write an explicit formula for

a_{n}

n^{\text {th }}

term of the sequence

20,23,26, \ldots

\newline

a_{n}=

Write an explicit formula for

a_{n}

n^{\text {th }}

term of the sequence

35,28,21, \ldots

\newline

a_{n}=

Write an explicit formula for

a_{n}

n^{\text {th }}

term of the sequence

6,2,-2, \ldots

\newline

a_{n}=

Write an explicit formula for

a_{n}

n^{\text {th }}

term of the sequence

37,31,25, \ldots

\newline

a_{n}=

Write an explicit formula for

a_{n}

n^{\text {th }}

term of the sequence

27,29,31, \ldots

\newline

a_{n}=

Write an explicit formula for

a_{n}

n^{\text {th }}

term of the sequence

150,30,6, \ldots

\newline

a_{n}=

Write an explicit formula for

a_{n}

n^{\text {th }}

term of the sequence

80,20,5, \ldots

\newline

a_{n}=

The derivative of the function

f

f^{\prime}(x)=x^{2}+3 \sin (3 x)

x

values, if any, in the interval

-2<x<2.5

where the function

f

has a relative minimum. You may use a calculator and round all values to

3

decimal places.

\newline

x=

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

g(x) = -(x-4)^{2} + 5

are graphed in the same

xy

y = f(x)

y = g(x)

(h_{1}, k_{1})

is the vertex of the graph of function

f

(h_{2}, k_{2})

is the vertex of the graph of function

g

k_{2} - k_{1}

\newline

1

\newline

-2

\newline

0

\newline

2

\newline

8

Triangle BCD is dilated by a scale factor of

\frac{2}{3}

to form triangle

B^{\prime} C^{\prime} D^{\prime}

B^{\prime} C^{\prime}

12

. What is the measure of side

\mathrm{BC}

\newline

A figure containing

\angle P Q R

is dilated by a scale factor of

\frac{1}{2}

to form a new figure which contains

\angle P^{\prime} Q^{\prime} R^{\prime} . \angle P Q R

107^{\circ}

. What is the measure of

\angle P^{\prime} Q^{\prime} R^{\prime}

\newline

\mathrm{m} \angle P^{\prime} Q^{\prime} R^{\prime}=\square^{\circ}

A figure containing

\angle E F G

is dilated by a scale factor of

2

to form a new figure which contains

\angle E^{\prime} F^{\prime} G^{\prime} . \angle E F G

59^{\circ}

. What is the measure of

\angle E^{\prime} F^{\prime} G^{\prime}

\newline

\mathrm{m} \angle E^{\prime} F^{\prime} G^{\prime}=\square^{\circ}

A figure containing

\angle D E F

is dilated by a scale factor of

5

to form a new figure which contains

\angle D^{\prime} E^{\prime} F^{\prime} . \angle D^{\prime} E^{\prime} F^{\prime}

36^{\circ}

. What is the measure of

\angle D E F

\newline

\mathrm{m} \angle D E F=

If the equation

y = x^{2} + 9

is graphed in the

xy

-plane, what are the coordinates of the vertex of the graph?

\newline

1

\newline

(-3,0)

\newline

(0,3)

\newline

(0,9)

\newline

(9,0)

For the function

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

, find the slope of the secant line between

x=1

x=3

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