If there is one prayer that you should

- Samuel Dominic Chukwuemeka
**pray/sing** every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)

It is the **most powerful prayer**.
A **pure heart**, a **clean mind**, and a **clear conscience** is necessary for it.

For in GOD we live, and move, and have our being.

- Acts 17:28

The

- Samuel Dominic Chukwuemeka**Joy** of a **Teacher** is the **Success** of his **Students.**

For ACT Students

The ACT is a timed exam...60 questions for 60 minutes

This implies that you have to solve each question in one minute.

Some questions will typically take less than a minute a solve.

Some questions will typically take more than a minute to solve.

The goal is to maximize your time. You use the time saved on those questions you
solved in less than a minute, to solve the questions that will take more than a minute.

So, you should try to solve each question __correctly__ and __timely__.

So, it is not just solving a question correctly, but solving it __correctly on time__.

Please ensure you attempt __all ACT questions__.

There is no *negative* penalty for a wrong answer.

Solve all questions

Show all work

(1.) **ACT** A point at (−5, 7) in the standard (*x, y*) coordinate plane is translated right 7 coordinate units and down 5 coordinate units.

What are the coordinates of the point after the translation?

$ Point: (-5, 7) \\[3ex] x = -5, y = 7 \\[3ex] HOSH\:\: 7 \:\:units\:\: right:\;\;x-coordinate \:\:changes \\[3ex] -5 + 7 = 2 \\[3ex] (-5, 7) \rightarrow (2, 7) \\[3ex] x = 2, y = 7 \\[3ex] VESH\:\: 5 \:\:units\:\: down:\;\; y-coordinate \:\:changes \\[3ex] 7 - 5 = 2 \\[3ex] (2, 7) \rightarrow (2, 2) $

What are the coordinates of the point after the translation?

$ Point: (-5, 7) \\[3ex] x = -5, y = 7 \\[3ex] HOSH\:\: 7 \:\:units\:\: right:\;\;x-coordinate \:\:changes \\[3ex] -5 + 7 = 2 \\[3ex] (-5, 7) \rightarrow (2, 7) \\[3ex] x = 2, y = 7 \\[3ex] VESH\:\: 5 \:\:units\:\: down:\;\; y-coordinate \:\:changes \\[3ex] 7 - 5 = 2 \\[3ex] (2, 7) \rightarrow (2, 2) $

(2.) Complete the underlined:

Suppose that the graph of a function*f* is known.

Then the graph of*y* = *f*(*x*−2) may be obtained by a .............. shift of the graph of *f* .............. a distance of 2 units.

Suppose that the graph of a function*f* is known.

Then the graph of*y* = *f*(*x*−2) may be obtained by a __horizontal__ shift of the graph of *f* __right__ a distance of 2 units.

Suppose that the graph of a function

Then the graph of

Suppose that the graph of a function

Then the graph of

(3.) Which of the following functions has a graph that is the graph of $y = \sqrt{x}$ shifted up 6 units?

$ A.\;\; y = \sqrt{x} - 6 \\[3ex] B.\;\; y = \sqrt{x + 6} \\[3ex] C.\;\; y = \sqrt{x} + 6 \\[3ex] D.\;\; y = \sqrt{x - 6} \\[3ex] $

$ y = \sqrt{x} .......Parent\;\;Function \\[3ex] VESH\;\;6\;\;units\;\;up:\;\;y = \sqrt{x} + 6 ......Child\;\;function \\[3ex] $

$ A.\;\; y = \sqrt{x} - 6 \\[3ex] B.\;\; y = \sqrt{x + 6} \\[3ex] C.\;\; y = \sqrt{x} + 6 \\[3ex] D.\;\; y = \sqrt{x - 6} \\[3ex] $

$ y = \sqrt{x} .......Parent\;\;Function \\[3ex] VESH\;\;6\;\;units\;\;up:\;\;y = \sqrt{x} + 6 ......Child\;\;function \\[3ex] $

(4.) **ACT** Point *A* is located at (3, 8) in the standard (*x, y*) coordinate plane.

What are the coordinates of*A'*, the image of *A* after it is reflected across the *y-*axis?

**A.** (3, −8)

**B.** (−3, −8)

**C.** (−3, 8)

**D.** (8, 3)

**E.** (−8, 3)

Reflection across the y-axis is Horizontal Reflection (HORE)

$ HORE:\;\;x-coordinate\;\;changes \\[3ex] A(3, 8) \rightarrow A'(-3, 8) $

What are the coordinates of

Reflection across the y-axis is Horizontal Reflection (HORE)

$ HORE:\;\;x-coordinate\;\;changes \\[3ex] A(3, 8) \rightarrow A'(-3, 8) $

(5.) **ACT** A triangle, △*ABC*, is reflected across the *x-*axis to have the image
△*A'B'C'* in the standard (*x, y*) coordinate plane: thus, *A* reflects to *A'*.

The coordinates of point*A* are (*c, d*).

What are the coordinates of point*A'* ?

**F.** (*c, −d*)

**G.** (*−c, d*)

**H.** (*−c, −d*)

**J.** (*d, c*)

**K.** Cannot be determined from the given information.

Reflection across the x-axis is Vertical Reflection (VERE)

$ VERE:\;\;y-coordinate\;\;changes \\[3ex] A(c, d) \rightarrow A'(c, -d) $

The coordinates of point

What are the coordinates of point

Reflection across the x-axis is Vertical Reflection (VERE)

$ VERE:\;\;y-coordinate\;\;changes \\[3ex] A(c, d) \rightarrow A'(c, -d) $

(6.) Fill in the blank:

Suppose that the graph of a function*f* is known.

Then the graph of*y* = *f*(−*x*) may be obtained by a reflection about the $\hspace{10mm}$ of the graph of the function *y* = *f*(*x*)

Suppose that the graph of a function*f* is known.

Then the graph of y =*f*(−*x*) may be obtained by a reflection about the *y*-axis of the graph of the function *y* = *f*(*x*)

Suppose that the graph of a function

Then the graph of

Suppose that the graph of a function

Then the graph of y =

(7.) Find the function that is finally graphed after the following transformations are applied to the graph of $y = \sqrt{x}$ in the order listed.

(a.) Shift down 3 units

(b.) Reflect about the*x*-axis

(c.) Reflect about the*y*-axis

$ Parent\;\;Function:\;\; y = \sqrt{x} \\[3ex] 1st\;\;Transformation:\;\;VESH\;\;8\;\;units\;\;down \\[3ex] y = \sqrt{x} - 3 \\[5ex] 2nd\;\;Transformation:\;\;VERE \\[3ex] y = -(\sqrt{x} - 3) \\[5ex] 3rd\;\;Transformation:\;\;HORE \\[3ex] y = -(\sqrt{-x} - 3) = -\sqrt{-x} + 3 \\[5ex] $

(a.) Shift down 3 units

(b.) Reflect about the

(c.) Reflect about the

$ Parent\;\;Function:\;\; y = \sqrt{x} \\[3ex] 1st\;\;Transformation:\;\;VESH\;\;8\;\;units\;\;down \\[3ex] y = \sqrt{x} - 3 \\[5ex] 2nd\;\;Transformation:\;\;VERE \\[3ex] y = -(\sqrt{x} - 3) \\[5ex] 3rd\;\;Transformation:\;\;HORE \\[3ex] y = -(\sqrt{-x} - 3) = -\sqrt{-x} + 3 \\[5ex] $

(8.) **ACT** In the standard (*x, y*) coordinate plane, *A'* is the image resulting from the
reflection of the point *A*(2, −3) across the *y*-axis.

What are the coordinates of*A'* ?

**A.** (−3, 2)

**B.** (−2, −3)

**C.** (−2, 3)

**D.** (2, 3)

**E.** (3, −2)

Reflection across the y-axis is Horizontal Reflection (HORE)

Only $x-value$ change

$y-value$ does not change

$ Point: (2, −3) \\[3ex] x = 2, y = −3 \\[3ex] HORE:\;\;x-coordinate\;\;changes \\[3ex] (2, −3) \rightarrow (−2, −3) $

What are the coordinates of

Reflection across the y-axis is Horizontal Reflection (HORE)

Only $x-value$ change

$y-value$ does not change

$ Point: (2, −3) \\[3ex] x = 2, y = −3 \\[3ex] HORE:\;\;x-coordinate\;\;changes \\[3ex] (2, −3) \rightarrow (−2, −3) $

(9.) **ACT** In the standard (*x, y*) coordinate plane, the coordinates of the *y-intercept* of the
graph of the function $y = f(x)$ are (0, −2).

What are the coordinates of the*y-intercept* of the graph of the function $y = f(x) - 3$?

$ Point: (0, -2) \\[3ex] x = 0, y = -2 \\[3ex] f(x) - 3 = VESH\:\: 3\:\: units\:\: down:\:\: y-coordinate\:\: changes \\[3ex] -2 - 3 = -5 \\[3ex] (0, -2) \rightarrow (0, -5) $

What are the coordinates of the

$ Point: (0, -2) \\[3ex] x = 0, y = -2 \\[3ex] f(x) - 3 = VESH\:\: 3\:\: units\:\: down:\:\: y-coordinate\:\: changes \\[3ex] -2 - 3 = -5 \\[3ex] (0, -2) \rightarrow (0, -5) $

(10.) True or False: The graph of $y = \dfrac{1}{9}g(x)$ is the graph of $y = g(x)$ compressed by a factor of 9.

$ y = g(x)...Parent\;\;Function \\[3ex] y = \dfrac{1}{9}g(x)...Child\;\;Function \\[5ex] $ VECO: This is true because:

(1.) The graph of the child function is obtained by multiplying each*y*-coordinate of the parent function
__and__

(2.) $0 \lt \dfrac{1}{9} \lt 1$

$ y = g(x)...Parent\;\;Function \\[3ex] y = \dfrac{1}{9}g(x)...Child\;\;Function \\[5ex] $ VECO: This is true because:

(1.) The graph of the child function is obtained by multiplying each

(2.) $0 \lt \dfrac{1}{9} \lt 1$

(11.) Write the function whose graph of:

(a.)*y* = *x*², is shifted to the right 6 units.

(b.)*y* = *x*³, is shifted up 9 units.

(c.) $y = 7\sqrt[4]{x}$, is reflected about the*x*-axis.

(d.)*y* = |*x*|, is vertically stretched by a factor of 6.

(a.) HOSH 6 units right:*y* = (*x* − 6)²

(b.) VESH 9 units up:*y* = *x*³ + 9

(c.) VERE: $y = -7\sqrt[4]{x}$

(d.) VEST by a factor of 6:*y* = 6|*x*|

(a.)

(b.)

(c.) $y = 7\sqrt[4]{x}$, is reflected about the

(d.)

(a.) HOSH 6 units right:

(b.) VESH 9 units up:

(c.) VERE: $y = -7\sqrt[4]{x}$

(d.) VEST by a factor of 6:

(12.) **ACT** In the standard (*x, y*) coordinate plane, point *A* has coordinate (−8, −3).

Point*A* is translated 8 units to the right and 3 units up, and that image is labeled *A'.*

What are the coordinates of*A'* ?

**A.** (−16, −6)

**B.** (−11, −11)

**C.** (−8, −6)

**D.** (0, 0)

**E.** (16, 6)

A(−8, −3)

HOSH 8 units right

A(−8, −3) → A'(−8 + 8, −3)

→ A'(0, −3)

VESH 3 units up

A'(0, −3) → A''(0, −3 + 3)

→ A''(0, 0)

Point

What are the coordinates of

A(−8, −3)

HOSH 8 units right

A(−8, −3) → A'(−8 + 8, −3)

→ A'(0, −3)

VESH 3 units up

A'(0, −3) → A''(0, −3 + 3)

→ A''(0, 0)

(13.) **ACT** In the standard (*x, y*) coordinate plane, given Parabola A with equation $y = 3x^2$, Parabola B is the image of Parabola A after a shift of 7 coordinate units
to the left and 4 coordinate units down.

Parabola B has which of the following equations?

$ F.\;\; y = 3(x - 4)^2 - 7 \\[3ex] G.\;\; y = 3(x - 7)^2 - 4 \\[3ex] H.\;\; y = 3(x - 7)^2 + 4 \\[3ex] J.\;\; y = 3(x + 7)^2 - 4 \\[3ex] K.\;\; y = 3(x + 7)^2 + 4 \\[3ex] $

Parabola A: Equation:*y* = 3*x*²

7 coordinate units to the left = HOSH 7 units left = 3(*x* + 7)²

4 coordinate units down = VESH 4 units down = 3(*x* + 7)² − 4

Parabola B: Equation:*y* = 3(*x* + 7)² − 4

The answer is Option**J.**

Parabola B has which of the following equations?

$ F.\;\; y = 3(x - 4)^2 - 7 \\[3ex] G.\;\; y = 3(x - 7)^2 - 4 \\[3ex] H.\;\; y = 3(x - 7)^2 + 4 \\[3ex] J.\;\; y = 3(x + 7)^2 - 4 \\[3ex] K.\;\; y = 3(x + 7)^2 + 4 \\[3ex] $

Parabola A: Equation:

7 coordinate units to the left = HOSH 7 units left = 3(

4 coordinate units down = VESH 4 units down = 3(

Parabola B: Equation:

The answer is Option

(14.) Fill in the blank:

Suppose that the graph of a function*f* is known.

Then the graph of*y* = −*f*(*x*) may be obtained by a reflection about the $\hspace{10mm}$ of the graph of the function *y* = *f*(*x*)

Suppose that the graph of a function*f* is known.

Then the graph of y =*f*(−*x*) may be obtained by a reflection about the *x*-axis of the graph of the function *y* = *f*(*x*)

Suppose that the graph of a function

Then the graph of

Suppose that the graph of a function

Then the graph of y =

(15.) Find the function that is finally graphed after the following transformations are applied to the graph of $y = \sqrt{x}$ in the order listed.

(a.) Reflect about the*x*-axis

(b.) Shift up 7 units

(c.) Shift right 5 units

$ Parent\;\;Function:\;\; y = \sqrt{x} \\[3ex] 1st\;\;Transformation:\;\;VERE \\[3ex] y = -\sqrt{x} \\[5ex] 2nd\;\;Transformation:\;\;VESH\;\;7\;\;units\;\;down \\[3ex] y = -\sqrt{x} + 7 \\[5ex] 3rd\;\;Transformation:\;\;HOSH\;\;5\;\;units\;\;right \\[3ex] y = -\sqrt{x - 5} + 7 \\[5ex] $

(a.) Reflect about the

(b.) Shift up 7 units

(c.) Shift right 5 units

$ Parent\;\;Function:\;\; y = \sqrt{x} \\[3ex] 1st\;\;Transformation:\;\;VERE \\[3ex] y = -\sqrt{x} \\[5ex] 2nd\;\;Transformation:\;\;VESH\;\;7\;\;units\;\;down \\[3ex] y = -\sqrt{x} + 7 \\[5ex] 3rd\;\;Transformation:\;\;HOSH\;\;5\;\;units\;\;right \\[3ex] y = -\sqrt{x - 5} + 7 \\[5ex] $

(16.) **ACT** The graph of *y* = |*x* − 6| is in the standard (*x, y*) coordinate plane.

Which of the following transformations, when applied to the graph of*y* = |*x*|, results in the graph of *y* = |*x* − 6|?

**F.** Translation to the right 6 coordinate units

**G.** Translation to the left 6 coordinate units

**H.** Translation up 6 coordinate units

**J.** Translation down 6 coordinate units

**K.** Reflection across the line *x* = 6

Parent Function:*y* = |*x*|

HOSH 6 units right

Transformed (Child) Function:*y* = |*x* − 6|

The answer is Option**F.:** Translation to the right 6 coordinate units

Which of the following transformations, when applied to the graph of

Parent Function:

HOSH 6 units right

Transformed (Child) Function:

The answer is Option

(17.) **ACT** A line segment has endpoints (*a, b*) and (*c, d*) in the standard (*x, y*) coordinate
plane, where *a, b, c*, and *d* are distinct positive integers.

The segment is reflected across the*x-*axis.

After this reflection, what are the coordinates of the endpoints of the image?

$ F.\;\; (-a, b)\;\;and\;\;(-c, d) \\[3ex] G.\;\; (a, -b)\;\;and\;\;(c, -d) \\[3ex] H.\;\; (-a, -b)\;\;and\;\;(-c, -d) \\[3ex] J.\;\; (a, b)\;\;and\;\;(c, d) \\[3ex] K.\;\; (a, 0)\;\;and\;\;(c, 0) \\[3ex] $

For any point, (*x, y*); reflection across the x-axis gives (*x*, −*y*)

This implies

$ (a, b) \rightarrow (a, -b) \\[3ex] (c, d) \rightarrow (c, -d) \\[5ex] Option\;G:\;\; (a, -b)\;\;and\;\;(c, -d) $

The segment is reflected across the

After this reflection, what are the coordinates of the endpoints of the image?

$ F.\;\; (-a, b)\;\;and\;\;(-c, d) \\[3ex] G.\;\; (a, -b)\;\;and\;\;(c, -d) \\[3ex] H.\;\; (-a, -b)\;\;and\;\;(-c, -d) \\[3ex] J.\;\; (a, b)\;\;and\;\;(c, d) \\[3ex] K.\;\; (a, 0)\;\;and\;\;(c, 0) \\[3ex] $

For any point, (

This implies

$ (a, b) \rightarrow (a, -b) \\[3ex] (c, d) \rightarrow (c, -d) \\[5ex] Option\;G:\;\; (a, -b)\;\;and\;\;(c, -d) $

(18.) Which of the following functions has a graph that is the graph of $y = f(x)$ stretched horizontally by a factor of 10?

$ A.\;\; y = \dfrac{1}{10}f(x) \\[5ex] B.\;\; y = 10f(x) \\[3ex] C.\;\; y = f\left(\dfrac{1}{10}x\right) \\[5ex] D.\;\; y = f(10x) \\[3ex] $

$ y = f(x) .......Parent\;\;Function \\[3ex] HOST\;\;by\;\;factor\;\;of\;\;10:\;\;y = f\left(\dfrac{1}{10}x\right) ......Child\;\;function $

$ A.\;\; y = \dfrac{1}{10}f(x) \\[5ex] B.\;\; y = 10f(x) \\[3ex] C.\;\; y = f\left(\dfrac{1}{10}x\right) \\[5ex] D.\;\; y = f(10x) \\[3ex] $

$ y = f(x) .......Parent\;\;Function \\[3ex] HOST\;\;by\;\;factor\;\;of\;\;10:\;\;y = f\left(\dfrac{1}{10}x\right) ......Child\;\;function $

(19.) **ACT** The point (3, 27) is labeled on the graph of *f*(*x*) = *x*³ in the standard (*x, y*) coordinate plane below.

The graph of*f*(*x*) will be translated 3 coordinate units to the left.

Which of the following points will be on the image of the graph after the translation?

**F.** (0, 27)

**G.** (3, 24)

**H.** (3, 27)

**J.** (3, 30)

**K.** (6, 27)

Based on the graph:

When we move the graph (green color) 3 units left, the point (3, 27) on the graph becomes (0, 27) on the transformed graph (blue color)

It is seen as moving from Point A to Point B

However, please note the equations of the parent function and the transformed function

$ Parent\;\;Function:\;\; y = x^3 \\[3ex] Transformed\;\;Function:\;\;HOSH\;\;3\;\;units\;\;left:\;\; y = (x + 3)^3 $

The graph of

Which of the following points will be on the image of the graph after the translation?

Based on the graph:

When we move the graph (green color) 3 units left, the point (3, 27) on the graph becomes (0, 27) on the transformed graph (blue color)

It is seen as moving from Point A to Point B

However, please note the equations of the parent function and the transformed function

$ Parent\;\;Function:\;\; y = x^3 \\[3ex] Transformed\;\;Function:\;\;HOSH\;\;3\;\;units\;\;left:\;\; y = (x + 3)^3 $

(20.) **ACT** In the standard (*x, y*) coordinate plane, the graph of the function $y = 5\sin(x) - 7$ undergoes a single translation such that the equation of
its image is $y = 5\sin(x) - 14$.

Which of the following describes this translation?

**F.** Up 7 coordinate units

**G.** Down 7 coordinate units

**H.** Left 7 coordinate units

**J.** Right 7 coordinate units

**K.** Right 14 coordinate units

From −7 to −14

How do you get from −7 to −14?

What will you add to −7 to get −14?

Let the thing be*p*

$ -7 + p = -14 \\[3ex] p = -14 + 7 \\[3ex] p = -7 \\[3ex] $ This represents a VESH of 7 units down

This means: Option**G.** Down 7 coordinate units

Which of the following describes this translation?

From −7 to −14

How do you get from −7 to −14?

What will you add to −7 to get −14?

Let the thing be

$ -7 + p = -14 \\[3ex] p = -14 + 7 \\[3ex] p = -7 \\[3ex] $ This represents a VESH of 7 units down

This means: Option

(21.) **ACT** In the standard (*x, y*) coordinate plane below, △*ABC* will be translated 10 units down and then the resulting image will be
reflected over the *y*-axis.

What will be the coordinates of the final image of*A* resulting from both transformations?

**A.** (−5, 9)

**B.** (−1, 9)

**C.** (1, −9)

**D.** (5, −10)

**E.** (5, −9)

A(−5, 1)

VESH 10 units down

A(−5, 1) → A'(−5, 1 − 10)

→ A'(−5, −9)

HORE (Reflected over the*y*-axis)

A'(−5, −9) → A''(5, −9)

What will be the coordinates of the final image of

A(−5, 1)

VESH 10 units down

A(−5, 1) → A'(−5, 1 − 10)

→ A'(−5, −9)

HORE (Reflected over the

A'(−5, −9) → A''(5, −9)

(22.) The point (−19, 9) is on the graph of *y* = *f*(*x*)

Complete the underlined. Write an ordered pair.

(a.) A point on the graph of*y* = *k*(*x*), where *k*(*x*) = −*f*(*x*) is ............

(b.) A point on the graph of*y* = *k*(*x*), where *k*(*x*) = *f*(−*x*) is ............

(c.) A point on the graph of*y* = *k*(*x*), where *k*(*x*) = *f*(*x* + 9) is ............

(d.) A point on the graph of*y* = *k*(*x*), where *k*(*x*) = *f*(*x*) − 1 is ............

(e.) A point on the graph of*y* = *k*(*x*), where *k*(*x*) = $\dfrac{1}{3}$*f*(*x*) is ............

(f.) A point on the graph of*y* = *k*(*x*), where *k*(*x*) = 6*f*(*x*) is ............

(−19, 9) on*y* = *f*(*x*)

(a.) −*f*(*x*) is VERE

(−19, 9) → (−19, −9)

(b.)*f*(−*x*) is HORE

(−19, 9) → [−(−19), 9]

→ (19, 9)

(c.)*f*(*x* + 9) is HOSH 9 units left

(−19, 9) → (−19 − 9, 9)

→ (−28, 9)

(d.)*f*(*x*) − 1 is VESH 1 unit down

(−19, 9) → (−19, 9 − 1)

→ (−19, 8)

(e.) $\dfrac{1}{3}$*f*(*x*) is VECO by a factor of $\dfrac{1}{3}$

$ (-19, 9) \rightarrow \left(-19, \dfrac{1}{3} * 9\right) \\[5ex] \hspace{8ex} \rightarrow (-19, 3) \\[3ex] $ (f.) 6*f*(*x*) is VEST by a factor of 6

(−19, 9) → (−19, 6 * 9)

→ (−19, 54)

Complete the underlined. Write an ordered pair.

(a.) A point on the graph of

(b.) A point on the graph of

(c.) A point on the graph of

(d.) A point on the graph of

(e.) A point on the graph of

(f.) A point on the graph of

(−19, 9) on

(a.) −

(−19, 9) → (−19, −9)

(b.)

(−19, 9) → [−(−19), 9]

→ (19, 9)

(c.)

(−19, 9) → (−19 − 9, 9)

→ (−28, 9)

(d.)

(−19, 9) → (−19, 9 − 1)

→ (−19, 8)

(e.) $\dfrac{1}{3}$

$ (-19, 9) \rightarrow \left(-19, \dfrac{1}{3} * 9\right) \\[5ex] \hspace{8ex} \rightarrow (-19, 3) \\[3ex] $ (f.) 6

(−19, 9) → (−19, 6 * 9)

→ (−19, 54)

(23.) Write the function whose graph is the graph of:

(a.) $y = 7\sqrt{x}$ is reflected about the*y*-axis

(b.) $y = x^3$, is horizontally stretched by a factor by 2.

$ (a.) \\[3ex] y = 7\sqrt{x} \\[3ex] Reflected\;\;about\;\;the\;\;y-axis = Horizontal\;\;Reflection \\[3ex] y = 7\sqrt{-x} \\[5ex] (b.) \\[3ex] y = x^3 \\[3ex] Horizontally\;\;stretched\;\;by\;\;a\;\;factor\;\;of\;\;2 \\[3ex] y = \left(\dfrac{1}{2}x\right)^3 \\[5ex] $

(a.) $y = 7\sqrt{x}$ is reflected about the

(b.) $y = x^3$, is horizontally stretched by a factor by 2.

$ (a.) \\[3ex] y = 7\sqrt{x} \\[3ex] Reflected\;\;about\;\;the\;\;y-axis = Horizontal\;\;Reflection \\[3ex] y = 7\sqrt{-x} \\[5ex] (b.) \\[3ex] y = x^3 \\[3ex] Horizontally\;\;stretched\;\;by\;\;a\;\;factor\;\;of\;\;2 \\[3ex] y = \left(\dfrac{1}{2}x\right)^3 \\[5ex] $

(24.) **ACT** Point *A* is located at (3, 8) in the standard (*x, y*) coordinate plane.

What are the coordinates of*A'*, the image of *A* after it is reflected across the *y-*axis?

$ A.\;\; (3, -8) \\[3ex] B.\;\; (-3, -8) \\[3ex] C.\;\; (-3, 8) \\[3ex] D.\;\; (8, 3) \\[3ex] E.\;\; (-8, 3) \\[3ex] $

For any point, (x, y); reflection across the y-axis gives (-x, y)

This implies

$ A(3, 8) \rightarrow A'(-3, 8) $

What are the coordinates of

$ A.\;\; (3, -8) \\[3ex] B.\;\; (-3, -8) \\[3ex] C.\;\; (-3, 8) \\[3ex] D.\;\; (8, 3) \\[3ex] E.\;\; (-8, 3) \\[3ex] $

For any point, (x, y); reflection across the y-axis gives (-x, y)

This implies

$ A(3, 8) \rightarrow A'(-3, 8) $

(25.) **ACT** The function *y* = *f*(*x*) is graphed in the standard (*x, y*) coordinate plane below.

The points on the graph of the function*y* = 3 + *f*(*x* − 1) can be obtained from the points on *y* = *f*(*x*) by a shift of:

**A.** 1 unit to the right and 3 units up.

**B.** 1 unit to the right and 3 units down.

**C.** 3 units to the right and 1 unit up.

**D.** 3 units to the right and 1 unit down.

**E.** 3 units to the left and 1 unit down.

Parent Function:*y* = *f*(*x*)

HOSH 1 unit right

Transformed (Child) Function:*y* = *f*(*x* − 1)

VESH 3 units up

Transformed Function:*y* = *f*(*x* − 1) + 3

*y* = 3 + *f*(*x* − 1)

The answer is Option**A.**: 1 unit to the right and 3 units up.

The points on the graph of the function

Parent Function:

HOSH 1 unit right

Transformed (Child) Function:

VESH 3 units up

Transformed Function:

The answer is Option

(26.) The graph of the function $f\left(\dfrac{1}{19}x\right)$ can be obtained from the graph of $y = f(x)$ by one of the following actions:

**A.** horizontally stretching the graph of *f*(*x*) by a factor 19

**B.** horizontally compressing the graph of *f*(*x*) by a factor 19

**C.** vertically stretching the graph of *f*(*x*) by a factor 19

**D.** vertically compressing the graph of *f*(*x*) by a factor 19

$f\left(\dfrac{1}{19}x\right)$ means that it is something "horizontally" because the $\dfrac{1}{19}$ is inside with the*x*

$a = \dfrac{1}{19}$

Because: $0 \lt \dfrac{1}{19} \lt 1$; it is horizontally stretched by a factor of $1 \div \dfrac{1}{19}$

So, it is horizontally stretched by a factor of 19.

$f\left(\dfrac{1}{19}x\right)$ means that it is something "horizontally" because the $\dfrac{1}{19}$ is inside with the

$a = \dfrac{1}{19}$

Because: $0 \lt \dfrac{1}{19} \lt 1$; it is horizontally stretched by a factor of $1 \div \dfrac{1}{19}$

So, it is horizontally stretched by a factor of 19.

(27.)

(28.) **ACT** In the standard (*x, y*) coordinate plane, point *A* has coordinates (−7, −5).

Point*A* is translated 7 units to the left and 5 units down, and that image is labeled *A'*.

What are the coordinates of*A'*?

$ F.\;\; (-14, -10) \\[3ex] G.\;\; (-12, -12) \\[3ex] H.\;\; (-7, -10) \\[3ex] J.\;\; (0, 0) \\[3ex] K.\;\; (14, 10) \\[3ex] $

Translated to the left or right is only for the x-coordinate

Translated up or down is only for the y-coordinate

$ \underline{Translated\;\;7\;\;units\;\;to\;\;the\;\;left} \\[3ex] A(-7, -5) \rightarrow A'(-7 - 7, -5) \\[3ex] \hspace{4.5em} \rightarrow A'(-14, -5) \\[3ex] \underline{Translated\;\;5\;\;units\;\;down} \\[3ex] A'(-14, -5) \rightarrow A''(-14, -5 - 5) \\[3ex] \hspace{5.2em} \rightarrow A''(-14, -10) $

Point

What are the coordinates of

$ F.\;\; (-14, -10) \\[3ex] G.\;\; (-12, -12) \\[3ex] H.\;\; (-7, -10) \\[3ex] J.\;\; (0, 0) \\[3ex] K.\;\; (14, 10) \\[3ex] $

Translated to the left or right is only for the x-coordinate

Translated up or down is only for the y-coordinate

$ \underline{Translated\;\;7\;\;units\;\;to\;\;the\;\;left} \\[3ex] A(-7, -5) \rightarrow A'(-7 - 7, -5) \\[3ex] \hspace{4.5em} \rightarrow A'(-14, -5) \\[3ex] \underline{Translated\;\;5\;\;units\;\;down} \\[3ex] A'(-14, -5) \rightarrow A''(-14, -5 - 5) \\[3ex] \hspace{5.2em} \rightarrow A''(-14, -10) $

(29.) **ACT** In the standard (*x, y*) coordinate plane below, $\triangle$*ABC* will be translated 10 units
down and then the resulting image will be reflected over the *y-*axis.

What will be the coordinates of the final image of*A* resulting from both transformations?

$ A.\;\; (-5, 9) \\[3ex] B.\;\; (-1, 9) \\[3ex] C.\;\; (1, -9) \\[3ex] D.\;\; (5, -10) \\[3ex] E.\;\; (5, -9) \\[3ex] $

Translated up or down is only for the*y-*coordinate

For any point, (x, y); reflection across the y-axis gives (-x, y)

We are only concerned with Point*A*

$ \underline{Translated\;\;10\;\;units\;\;down} \\[3ex] A(-5, 1) \rightarrow A'(-5, 1 - 10) \\[3ex] \hspace{3.7em} \rightarrow A'(-5, -9) \\[5ex] \underline{Reflected\;\;over\;\;y-axis} \\[3ex] A'(-5, -9) \rightarrow A''(5, -9) $

What will be the coordinates of the final image of

$ A.\;\; (-5, 9) \\[3ex] B.\;\; (-1, 9) \\[3ex] C.\;\; (1, -9) \\[3ex] D.\;\; (5, -10) \\[3ex] E.\;\; (5, -9) \\[3ex] $

Translated up or down is only for the

For any point, (x, y); reflection across the y-axis gives (-x, y)

We are only concerned with Point

$ \underline{Translated\;\;10\;\;units\;\;down} \\[3ex] A(-5, 1) \rightarrow A'(-5, 1 - 10) \\[3ex] \hspace{3.7em} \rightarrow A'(-5, -9) \\[5ex] \underline{Reflected\;\;over\;\;y-axis} \\[3ex] A'(-5, -9) \rightarrow A''(5, -9) $

(30.) Which transformations are needed to graph the function $f(x) = 3(x + 1)^2 - 2$ ?

Choose the correct answer below.

**A.** The graph of *y* = *x*² should be horizontally shifted to the left by 1 unit, vertically stretched by a factor of 3, and shifted vertically up by 2 units.

**B.** The graph of *y* = *x*² should be horizontally shifted to the right by 1 unit, vertically compressed by a factor of 3, and shifted vertically up by 2 units.

**C.** The graph of *y* = *x*² should be horizontally shifted to the left by 1 unit, vertically stretched by a factor of 3, and shifted vertically down by 2 units.

**D.** The graph of *y* = *x*² should be horizontally shifted to the right by 1 unit, vertically compressed by a factor of 3, and shifted vertically down by 2 units.

$f(x) = 3(x + 1)^2 - 2$

Parent Function:*f*(*x*) = *x*²

(*x* + 1)²: HOSH 1 unit left

3(*x* + 1)²: VEST by a factor of 3

3(*x* + 1)² - 2: VESH 2 units down

The correct option is**C.**

Choose the correct answer below.

$f(x) = 3(x + 1)^2 - 2$

Parent Function:

(

3(

3(

The correct option is

(31.)

(32.) **ACT** A point at (−3, 7) in the standard (*x, y*) coordinate plane is shifted down 3 units and right 7 units.

What are the coordinates of the new point?

$ F.\;\; (-10, 10) \\[3ex] G.\;\; (0, 0) \\[3ex] H.\;\; (4, 4) \\[3ex] J.\;\; (4, 10) \\[3ex] K.\;\; (10, 10) \\[3ex] $

Let the point be*A*(-3, 7)

shifted down: only the y-coordinate is affected

shifted right: only the x-coordinate is affected

$ \underline{shifted\;\;down\;\;3\;\;units} \\[3ex] A(-3, 7) \rightarrow A'(-3, 7 - 3) \\[3ex] \hspace{4em} \rightarrow A'(-3, 4) \\[5ex] \underline{shifted\;\;right\;\;7\;\;units} \\[3ex] A'(-3, 4) \rightarrow A''(-3 + 7, 4) \\[3ex] \hspace{4em} \rightarrow A''(4, 4) $

What are the coordinates of the new point?

$ F.\;\; (-10, 10) \\[3ex] G.\;\; (0, 0) \\[3ex] H.\;\; (4, 4) \\[3ex] J.\;\; (4, 10) \\[3ex] K.\;\; (10, 10) \\[3ex] $

Let the point be

shifted down: only the y-coordinate is affected

shifted right: only the x-coordinate is affected

$ \underline{shifted\;\;down\;\;3\;\;units} \\[3ex] A(-3, 7) \rightarrow A'(-3, 7 - 3) \\[3ex] \hspace{4em} \rightarrow A'(-3, 4) \\[5ex] \underline{shifted\;\;right\;\;7\;\;units} \\[3ex] A'(-3, 4) \rightarrow A''(-3 + 7, 4) \\[3ex] \hspace{4em} \rightarrow A''(4, 4) $

(33.) **ACT** A point with coordinates (*a, b*) is plotted in the standard (*x, y*) coordinate plane as
shown below.

The point is then reflected across the*y-*axis.

Which of the following are the coordinates for the point after the reflection?

$ A.\;\; (-a, b) \\[3ex] B.\;\; (a, -b) \\[3ex] C.\;\; (b, a) \\[3ex] D.\;\; (-b, a) \\[3ex] E.\;\; (b, -a) \\[3ex] $

For any point, (x, y); reflection across the y-axis gives (-x, y)

This implies

$ (a, b) \rightarrow (-a, b) $

The point is then reflected across the

Which of the following are the coordinates for the point after the reflection?

$ A.\;\; (-a, b) \\[3ex] B.\;\; (a, -b) \\[3ex] C.\;\; (b, a) \\[3ex] D.\;\; (-b, a) \\[3ex] E.\;\; (b, -a) \\[3ex] $

For any point, (x, y); reflection across the y-axis gives (-x, y)

This implies

$ (a, b) \rightarrow (-a, b) $

(34.)

(35.)

(36.) **ACT** A point at (−2, 8) in the standard (*x, y*) coordinate plane is shifted right 8 units and down
2 units.

What are the coordinates of the new point?

$ F.\;\; (-10, 10) \\[3ex] G.\;\; (0, 0) \\[3ex] H.\;\; (6, 6) \\[3ex] J.\;\; (6, 10) \\[3ex] K.\;\; (10, 10) \\[3ex] $

Let the point be*A*(-2, 8)

shifted right: only the x-coordinate is affected

shifted down: only the y-coordinate is affected

$ \underline{shifted\;\;right\;\;8\;\;units} \\[3ex] A(-2, 8) \rightarrow A'(-2 + 8, 8) \\[3ex] \hspace{4em} \rightarrow A'(6, 8) \\[3ex] \underline{shifted\;\;down\;\;2\;\;units} \\[3ex] A'(6, 8) \rightarrow A''(6, 8 - 2) \\[3ex] \hspace{3.5em} \rightarrow A''(6, 6) $

What are the coordinates of the new point?

$ F.\;\; (-10, 10) \\[3ex] G.\;\; (0, 0) \\[3ex] H.\;\; (6, 6) \\[3ex] J.\;\; (6, 10) \\[3ex] K.\;\; (10, 10) \\[3ex] $

Let the point be

shifted right: only the x-coordinate is affected

shifted down: only the y-coordinate is affected

$ \underline{shifted\;\;right\;\;8\;\;units} \\[3ex] A(-2, 8) \rightarrow A'(-2 + 8, 8) \\[3ex] \hspace{4em} \rightarrow A'(6, 8) \\[3ex] \underline{shifted\;\;down\;\;2\;\;units} \\[3ex] A'(6, 8) \rightarrow A''(6, 8 - 2) \\[3ex] \hspace{3.5em} \rightarrow A''(6, 6) $

(37.)

(38.)

(39.)

(40.)

(41.)

(42.)

(43.)

(44.)

(45.)

(46.)