Question

The graph of ?(?) = 5 ^−? is reflected about the y-axis and compressed vertically by a factor of 1/3. Determine the equation of the new function, ?(?). Also state the y-intercept, domain, and range of ?(?).

Answer 1

The general rules for graphical transformation is:- Let y=f(x) be the original graph and g(x)=y'=a.f(bx+c)+d is the transformed graph, then Amplitude of y' is a times that of y/The graph of y is compressed/expanded vertically by factor of a. If a=-1, the graph is reflected about x axis The graph of y is compressed/expanded horizontally by factor of b. If b=-1, the graph is reflected about y axis. The graph of y is shifted horizontally/Phase shift by c units. If c is positive then left shift and vice versa. The graph of y is shifted Vertically by d units. If d is positive then upward shift and vice versa. Here, y=f(x)=5-x a=1/3 b=-1 Hence, g(x)=y=a.f(bx+c)+d

For y intercept, put x=0 => g(x)=50/3=1/3 y intercept is (0,1/3)

Domain of exponential function is the set of all real numbers. Hence x=R is the required domain.

Now, as x→ -∞, g(x)→ 0 and x → ∞, g(x) → ∞ and g(x) is continuously increasing. Hence range is (0,∞)

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