If the initial points of a parent function are... (description)

Question

 If the initial points of a parent function are (1,0) and (b,1) what are the new points if you perform the following transformations?

y = 2 log₃ (x + 1) - 3

Answer 1

The point (1,0) is on the graph of y = log₃x. If the point (b,1) is also on the graph, then log₃b = 1.

So, b = 3¹ = 3. Thus, (b,1) = (3,1).

To get the graph of y = 2log₃(x+1) - 3 from the graph of y = log₃x:

Step 1: Translate the graph of y = log₃x one unit to the left to get y = log₃(x+1). So the points (1,0) and (3,1) move to (0,0) and (2,1) respectively.

Step 2: Stretch y = log₃(x+1) vertically by a factor of 2 to get y = 2log₃(x+1). That is multiply the y-coordinates by 2. So, (0,0) stays the same and (2,1) moves to (2,2).

Step 3: Translate y = 2log₃(x+1) 3 units downward to get y = 2log₃(x+1) - 3. So, the point (0,0) moves to (0,-3) and (2,2) moves to (2,-5).

Answer: (1,0) → (0,-3) and ((b,1) → (2,-5) [→ means "moves to]