Question

Stretching, an(d)/(o)r reflecting. Find the function. g(x)=(x+3)^(3)-1

Answer 1

The given function
`\( g(x) = (x + 3)^3 - 1 \)` represents a cubic function with a horizontal shift to the left by 3 units and a vertical shift downward by 1 unit compared to the standard cubic function
`\( f(x) = x^3 \).`

Step-by-step explanation:

The general form of a cubic function is
`\( f(x) = x^3 \),` which has a single stationary point (inflection point) at the origin (0, 0). In this case, the given function
`\( g(x) = (x + 3)^3 - 1 \)` indicates transformations from the standard cubic function.

The term
`\( (x + 3) \)` inside the parentheses causes a horizontal shift to the left by 3 units. This means that the inflection point of the cubic function is moved from (0, 0) to (-3, 0). The term
`\( -1 \)` outside the parentheses represents a vertical shift downward by 1 unit, shifting the entire graph downward.

In summary, the function
`\( g(x) \)` is obtained by taking the standard cubic function and shifting it left by 3 units and down by 1 unit. The graph of \( g(x) \) will retain the basic shape of a cubic function but will be translated in the horizontal and vertical directions.