Question

Graph the function by starting with a function from the library of functions and then using the techniques of shifting, compressing, stretching, and/or reflecting. g(x) = x²-2.

Answer 1

Start with the function f(x) = x² and apply a vertical shift of 2 units downwards to graph the function g(x) = x² - 2.

Step-by-step explanation:

The function g(x) = x² - 2 can be derived by transforming the function f(x) = x² through a vertical shift downwards by 2 units. The base function f(x) = x² is a parabola centered at the origin, with its vertex at (0,0) and symmetric along the y-axis. Applying a vertical shift downwards shifts the entire graph of f(x) = x² downwards by 2 units.

The process involves taking the points on the graph of f(x) = x² and shifting them downward by 2 units to obtain the graph of g(x) = x² - 2. For example, the point (0,0) on the graph of f(x) = x² would shift to (0,-2) on the graph of g(x) = x² - 2. Similarly, other points on the graph of f(x) = x² would shift vertically downwards by 2 units to create the graph of g(x) = x² - 2.

The resulting graph of g(x) = x² - 2 will maintain the same shape as the graph of f(x) = x² but will be shifted downward by 2 units. It will still be a parabola opening upwards but will be translated downward compared to the original function f(x) = x². This downward shift represents the effect of the constant term (-2) in the function g(x) = x² - 2 on its graph, showcasing the impact of vertical transformations on functions.