the graph of a function f is illustrated to the right. use the graph of f as the first step toward graphing…
the graph of a function f is illustrated to the right. use the graph of f as the first step toward graphing each of the following functions. (a) ( f(x)=f(x)+3 ) (b) ( g(x)=f(x + 2) ) (c) ( p(x)=-f(x) ) (d) ( h(x)=f(x + 2)-2 ) (e) ( q(x)=\frac{1}{2} f(x) ) (f) ( g(x)=f(-x) ) (g) ( h(x)=f(2x) ) (g) choose the correct graph of ( h(x)=f(2x) ) below.
Answer
Explanation:
Step1: Recall the horizontal compression rule
For a function (y = f(kx)), if (k>1), the graph of (y = f(x)) is horizontally compressed by a factor of (\frac{1}{k}). Here (k = 2), so the graph of (y=f(x)) is horizontally compressed by a factor of (\frac{1}{2}).
Step2: Analyze the key - points
The original function (y = f(x)) has key - points at (x=-6\pi), (x = - 3\pi), (x=0), (x = 3\pi), (x=6\pi). For the function (y=f(2x)), when (2x=-6\pi), (x=-3\pi); when (2x=-3\pi), (x =-\frac{3\pi}{2}); when (2x = 0), (x = 0); when (2x=3\pi), (x=\frac{3\pi}{2}); when (2x = 6\pi), (x = 3\pi). The (y) - values of the function (y = f(2x)) are the same as the (y) - values of (y = f(x)) for the corresponding (x) - values.
Step3: Eliminate wrong options
Option A has a vertical compression (since the (y) - values are halved), which is for (y=\frac{1}{2}f(x)). Option B has a horizontal shift (since the (x) - values of key - points are not related by (x\to2x)). Option C has a vertical stretch (since the (y) - values are doubled). Option D: The graph of (y = f(x)) with (x) - values of key - points (x=-6\pi), (x=-3\pi), (x = 0), (x = 3\pi), (x=6\pi) is transformed to (x=-3\pi), (x=-\frac{3\pi}{2}), (x = 0), (x=\frac{3\pi}{2}), (x = 3\pi) (horizontal compression by a factor of (\frac{1}{2})) while keeping the (y) - values the same as in (y = f(x)).
Answer:
D.