Final answer:
The function g is obtained by applying a horizontal shrink by a factor of 2/3, followed by a translation 5 units left and 2 units down to f(x). The rule for g is g(x) = (9/4)(x+5)² - 2.
Step-by-step explanation:
To find a rule for g, which is a transformation of the function f(x) = x², we need to apply the given transformations in the correct order. First, we perform a horizontal shrink by a factor of 2/3, and then we apply a translation of 5 units left and 2 units down.
A horizontal shrink by a factor of 2/3 to the function f(x) can be accomplished by multiplying the input variable by the reciprocal of the shrink factor. This gives us f(3x/2). Next, a translation 5 units left is achieved by adding 5 to the input variable, resulting in f(3(x+5)/2). Finally, translating the graph 2 units down, we subtract 2 from the function, yielding g(x) = f(3(x+5)/2) - 2.
Substituting f(x) = x² into our expression for g(x), we get g(x) = (3(x+5)/2)² - 2. Simplifying that expression, we obtain the rule for g: g(x) = (9/4)(x+5)² - 2.