The linear functions f(x) and g(x) are represented on the graph, where g(x) is a transformation of f(x): A graph with two linear functions; f of x passes through 5, 0 and 10, 10, and g of x passes through negative 3, 0 and 2, 10. Part A: Describe two types of transformations that can be used to transform f(x) to g(x). (2 points) Part B: Solve for k in each type of transformation. (4 points) Part C: Write an equation for each type of transformation that can be used to transform f(x) to g(x). (4 points)

First, let's find the equations for our lines:

A linear relationship can be written as:

y = a*x + b

where a is the slope and b is the y-axis intercept.

For a line that passes through the points (x1, y1) and (x2, y2), the slope can be written as:

a = (y2 - y1)/(x2 - x1).

f(x) passes through (5,0) and (10, 10), then the slope is:

a = (10 - 0)/(10 - 5) = 10/5 = 2.

then we have:

y = 2*x + b

And when x = 5, we have y = 0.

0 = 2*5 + b

0 = 10 + b

b = -10

Then the equation for f(x) is:

y = f(x) = 2*x - 10.

Now for g(x) we have the points:

(3, 0) and (2, 10)

a = (10 - 0)/(2 - 3) = -10

y = -10*x + b

0 = -10*3 + b

b = 30.

y = g(x) = -10*x + 30.

A) Ok, the transformations:

Transformation 1 or T1.

f(x) = 2*x - 10

g(x) = -10*x + 30.

Then, we start with f(x):

First, we can move f(x) up 4 units and get:

f'(x) = 2*X - 6

Now we can dilate f(x) with a scale factor of -5 from the origin, now we get:

f''(x) = -5*f'(x) = -10*x + 30.

And this is g(x).

Transformation 2 or T2.

Move f(x) up 10 units, so now we have:

f'(x) = 2*x

Do a reflection over the x-axis, so the sign of y changes, and now we get:

f''(x) = -2*x

Do a dilation of scale factor 5

f'''(x) = 5*-2*x = -10*x

Now do a vertical translation of 30 units up.

f(x) = -10*x + 30 = g(x).

These are two transformations that start with f(x) and end with g(x).

B) Ok, as i was writting the transformations i already solved them, so this part is already done.

C) the equation for the transformations are:

T1) g(x) = -5*(f(x) + 4)

T2) g(x) =  -(f(x) + 10)*5 + 30

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab  in the y coordinate 

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

  x coordinate 

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

  y coordinate 

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

  x  coordinate 

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

  y  coordinate 

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

  x  coordinate 

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

  y  coordinate 

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

  x  coordinate 

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

  y  coordinate 

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

  x  coordinate 

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

  y  coordinate 

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

  x  coordinate 

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

  y  coordinate 

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

  x  coordinate 

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

  y  coordinate 

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕