The linear function f(x) and g(x) are represented on the graph where g(x) is a transformation of f(x):
sorry i messed up, this is the real one

part a:

6 units left is an option

18 units up is an option

part b:

i'm not too sure what they are calling k.

but if i had to guess it would be the k's in y=3(x-k)-1   and y=3x-1+k this case.

the first k i describe is -6 in this case.

the second k i describe is 18 in this case.

part c:

g(x)=3(x+6)-1

g(x)=(3x-1)+18

where f(x)=3x-1.

g(x)=f(x+6) moves the graph left 6 units (this is the first equation i wrote).

g(x)=f(x)+18 moves the graph up 18 units (this is the second equation i wrote).

i think this is what you are looking for.

step-by-step explanation:

part a:

ok these lines are parallel so we just have to figure out how much left/right to move f onto g.

or, we can figure out how much down/up to move f onto g.

let's look at a place where both lines cross nicely on the same horizontal line.

y=-1 looks nice.   i'm talking about the horizontal line that goes through the y-axis at -1 and has all coordinates in the form (x,-1).

count horizontally from (0,-1) on f to (-6,-1) on g, the direction was left and the number units we crossed was 6.

so f could be moved 6 units left to be transformed onto g.

now instead of looking horizontally, let's look vertically.

again we are looking for when it crosses nicely on a vertical line.   x=-1 looks good.   i'm talking about the vertical line that goes through the x-axis at x=-1 and whose coordinates are of the form (-1,y).

so i see (-1,-4) on f and (-1,14) on g. we are trying to figure out how to get (-1,-4) onto (-1,14), we have to move up 18 units. that is what you get if you count vertically from y=-4 to y=14.

part b and c:

we need the equation for f.

the slope-intercept form of a line is y=mx+b where m is the slope and b is the y-intercept.

b is where the line crosses the y-axis which for our graph of f is at (0,-1).

so b=-1.

now we are going to find the slope.

we identified two points on f already which were (-1,4) and (0,-1).

to find the slope i'm going to line them up vertically and subtract, then put 2nd difference over 1st difference.

(-1   ,-4)

-(0   ,-1)

-1     -3

so the slope is -3/-1 or just 3.

so the equation for f is y=3x-1.

now the value for k?

"from (0,-1) on f to (-6,-1) on g" is a quote from above to with this problem.

so here this effected x.

how do we get from the x's on f to x's on g?

we subtract 6 on the x's (from f to g):

so to do this for x's we must do the operation that is the inverse; addition.

y=3(x+6)-1

so k=-6 since the transformation that effects the x's is

(g(x)=f(x-k)=f(x+6) )

now the other value for k?

"(-1,-4) on f and (-1,14) on g" is another quote from above that will us.

this is effecting the y's.

we had to add 18 to -4 to get 14.

y=(3x-1)+18

the value for k here is 18.

(g(x)=f(x)+k=f(x)+18 )

step-by-step explanation: