Please help me with this

Question

The graph of f(x) contains the point (-5,-2).

(a) What point must be on the graph of f(x)+1?

(b) What point must be on the graph of f(x+1)?

(c) What point must be on the graph of f(x-4)-3?

Answer 1

(a) What point must be on the graph of f(x)+1? This is a vertical shift up 1 unit, so the y coordinate would change by 1 (-5,-2+1) = (-5,-1)

(b) What point must be on the graph of f(x+1)? This is a horizontal shift left 1, so the x coordinate would change by -1 unit (-5-1,-2) = (-6,-2)

(c) What point must be on the graph of f(x-4)-3? This is a vertical shift down 3 units, so the y coordinate would change by -3. It is also a horizontal shift right 4, so the x coordinate would change by 4 units. (-5+4,-2-3) = (-1,-5)

Answer 2

It seems as though you are working with translations of functions.

A horizontal translation (in the direction of the x-axis) can be described by:

g(x)= f(x-k), where k is the amount of units the function will translate in the horizontal direction (when k is positive, the translation will be to the right, when k is negative, the translation will be to the left)

A vertical translation (in the direction of the y-axis) can be described by

g(x)=f(x)+q, where q is the amount of units the function will translate in the vertical direction (when q is positive, the translation will be upwards, when q is negative, the translation will be downwards)

You are given that the point (-5,-2) is in the function f(x).

a) What point must be on the graph of f(x)+1?

Since the expression is of the form f(x)+q, with q=1, we can deduce that there will be a vertical translation upwards of 1 unit.

Answer: (-5,-1)

b) What point must be on the graph of f(x+1)?

Since the expression is of the form f(x-k), with k=-1, we can deduce that there will be a horizontal translation to the left of 1 unit.

Answer: (-6,-2)

c) What point must be on the graph of f(x-4)+3?

This expression has components of both a horizontal translation and a vertical translation.

For the vertical component, q= 3, meaning a vertical translation of 3 units upwards

For the horizontal component, k=4, meaning a horizontal translation of 4 units to the right

Answer: (-1,1)