The diagram shows part of the curve with equation y = f(x).

Question

The diagram shows part of the curve with equation y = f(x). The coordinates of the minimum point of this curve are (3,1). Write down the coordinates of the minimum point of the curve with equation

(a) y = f(x) + 3

(b) y = f(x - 2)

(c) y = f(1 and a half x)

Answer 1

Daniel,

 

I don't see your diagram, but it doesn't matter.

You say the minimum of y = f(x) occurs at (x,y) = (3,1). Hence y= f(x) >=1 for all other x in its domain of definition.

 

(a) y = f(x) + 3 merely increases the value y by 3 for any x, or just raises its graph up 3. The x-value for its minimum remains at x=3 and y = f(3) + 3 = 1 + 3 = 4.  Hence, new minimum = (3,4)

 

(b) y = f(x-2) just shifts or translates the graph of f to the right by 2.  So what occurred at x=3 (the minimum y) now occurs at x=5  without changing y = 1.  Hence, new minimum = (5,1).

 

(c) y = f(1.5x) compresses the x-scale without changing y.  So what happened at x=3 before now happens at x'=2 on the new scale. Thus, the minimum point now becomes  (2,1).