Question

WILL MARK BRAINIEST!! 50 POINTS.

Answer 1

Answer:

Part A: The two types of types of transformation are

1) Rotation of 11.3° about (1, 2)

2) By algebraic transformation

Part B:

Rotation by 11.3° and T(2 - y)×1/2 + x, 0)

Part C: The transformation that can be used to transform f(x) to g(x) is T(2 - y)×1/2 + x, 0)

Step-by-step explanation:

The coordinates through which the linear function f(x) passes = (1. 3) and (3, 13)

The coordinates through which the linear function g(x) passes = (1, 3) and (1, 13)

The equation for f(x) in slope and intercept form. y = m·x + c is given as follows;

The slope, m = (13 - 3)/(3 - 1) = 5

The equation in point and slope form is y - 3 = 5×(x -1)

y = 5·x - 5 + 3 = 5·x - 3

y = 5·x - 3

The equation for g(x) in slope and intercept form. y = m·x + c is given as follows;

The slope, m = (13 - 3)/(1 - 1) = ∞

∴ The equation in point and slope form is x = 1

Therefore, the two equations meet at the point (1, 2)

The transformation that can be used to transform f(x) to g(x) is T(2 - y)×1/2 + x, 0)

2) Another transformation that can be used is to rotate f(x) by the vertex angle as follows

Vertex angle is 90° - tan⁻¹(m) = 90° - tan⁻¹(5) ≈ 11.3°

Rotation of f(x) by 11.3° about (1, 2) gives g(x)

Hope this helped!

Answer 2

Answer:

Step-by-step explanation:

One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function  

g

(

x

)

=

f

(

x

)

+

k

, the function  

f

(

x

)

 is shifted vertically  

k

 units.

Graph of f of x equals the cubed root of x shifted upward one unit, the resulting graph passes through the point (0,1) instead of (0,0), (1, 2) instead of (1,1) and (-1, 0) instead of (-1, -1)

Figure 2. Vertical shift by  

k

=

1

 of the cube root function  

f

(

x

)

=

3

√

x

.

To help you visualize the concept of a vertical shift, consider that  

y

=

f

(

x

)

. Therefore,  

f

(

x

)

+

k

 is equivalent to  

y

+

k

. Every unit of  

y

 is replaced by  

y

+

k

, so the  

y

-

 value increases or decreases depending on the value of  

k

. The result is a shift upward or downward.

A GENERAL NOTE: VERTICAL SHIFT

Given a function  

f

(

x

)

, a new function  

g

(

x

)

=

f

(

x

)

+

k

, where  

k

 is a constant, is a vertical shift of the function  

f

(

x

)

. All the output values change by  

k

 units. If  

k

 is positive, the graph will shift up. If  

k

 is negative, the graph will shift down.

EXAMPLE 1: ADDING A CONSTANT TO A FUNCTION

To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 2 shows the area of open vents  

V

 (in square feet) throughout the day in hours after midnight,  

t

. During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.

Solution

HOW TO: GIVEN A TABULAR FUNCTION, CREATE A NEW ROW TO REPRESENT A VERTICAL SHIFT.

Identify the output row or column.

Determine the magnitude of the shift.

Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.

EXAMPLE 2: SHIFTING A TABULAR FUNCTION VERTICALLY

A function  

f

(

x

)

 is given below. Create a table for the function  

g

(

x

)

=

f

(

x

)

−

3

.

x

 2 4 6 8

f

(

x

)

 1 3 7 11

Show Solution

Identifying Horizontal Shifts

We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at h