The linear functions f(x) and g(x) are represented on the graph, where g(x) is a transformation of f(x):
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)

Part A) shift left 6 units or shift up 18 units

Part B) k=-6 and k=18

Part C) g(x)=f(x+6)  or g(x)=f(x)+18

Step-by-step explanation:

Find the equation of f(x)

we have the points(0,-1) and (2,5)

The slope m is

The equation in slope intercept form is

we have

substitute

Find the equation of g(x)

we have the points (-1,14) and (0,17)

The slope m is

The equation in slope intercept form is

we have

substitute

Part A and Part B: Describe two types of transformations that can be used to transform f(x) to g(x). Solve for k in each type of transformation

First type of transformation

we have

Find the x-intercepts of f(x) and g(x) (values of x when the value of the function is equal to zero)

For f(x)=0

For g(x)=0

f(x) -----> g(x)

The translation of the x-intercept is

(1/3,0) --------> (-17/3,0)

so

The rule of the translation is

(x,y) -----> (x+k,y)

(1/3,0) -----> (1/3+k,0)

so

(1/3+k,0)=(-17/3,0)

Solve for k

Multiply by 3 both sides

therefore

The rule of the translation is

(x,y) -----> (x-6,y)

That means -----> The translation is 6 units at left

Second type of transformation

we have

The y-intercepts (values of the function when the value of x is equal to zero)

f(x) -----> (0,-1)

g(x) ----> (0,17)

The translation of the y-intercept is

f(x) ---->g(x)

(0,-1) --------> (0,17)

so

The rule of the translation is

(x,y) -----> (x,y+k)

(0,-1) -----> (0,-1+k)

so

(0,-1+k)=(0,17)

Solve for k

therefore

The rule of the translation is

(x,y) -----> (x,y+18)

That means -----> The translation is 18 units up

Part C) Write an equation for each type of transformation that can be used to transform f(x) to g(x).

First transformation

The rule of the translation is

(x,y) -----> (x-6,y)

f(x) -----> g(x)

g(x)=f(x+6)

so

therefore

----> is correct

Second transformation

The rule of the translation is

(x,y) -----> (x,y+18)

f(x) -----> g(x)

so

----> is correct

the 3rd option is right

step-by-step explanation:

= 5

step-by-step explanation:

to find the 15th term substitute n = 15 into the nth term formula

= (- 3 × 15 ) + 50 = - 45 + 50 = 5