Write a rule for g. Let the graph of g be a horizontal stretch by a factor of 4, followed by a translation 3 units right and 5 units down of the graph of f(x)=x^(5)+3x

Question

Write a rule for g.

Let the graph of g be a horizontal stretch by a factor of 4, followed by a translation 3 units right and 5 units down of the graph of f(x)=
`x^(5)+3x`

Answer 1

To find the rule for g, apply a horizontal stretch by replacing x with x/4, then translate right by replacing x with (x-3) and down by subtracting 5, resulting in g(x)=[((x-3)/4)^5 + 3((x-3)/4)] - 5.

Step-by-step explanation:

To write a rule for g based on the given transformations of the function f(x)=x^5+3x, we need to apply a horizontal stretch by a factor of 4, and then translate the function 3 units to the right and 5 units down.

The horizontal stretch by a factor of 4 is applied by transforming the input variable x to x/4. This is because a horizontal stretch by a factor of k is achieved by replacing x with x/k. After applying the horizontal stretch, the function becomes f(\frac{x}{4})=(\frac{x}{4})^5 + 3(\frac{x}{4}).

Next, we translate the stretched graph 3 units to the right by replacing x with (x-3), and then 5 units down which is achieved by subtracting 5 from the entire function. Therefore, the rule for g becomes g(x)=[(\frac{x-3}{4})^5 + 3(\frac{x-3}{4})] - 5.