Homework Buster- Math

The transformation f(x)↦f(x)–10 shifts the graph of f(x) downward by 10 units. To see why, let's first recall what the function f(x) represents. In general, a function f(x) takes an input value x and produces an output value y = f(x). In the context of graphing, we typically plot the points (x, f(x)) on a coordinate plane to create the graph of the function. This graph represents the relationship between the input values x and the output values f(x). Now let's consider what happens when we apply the transformation f(x)↦f(x)–10 to the function f(x). This transformation simply subtracts 10 from the output value f(x) for each input value x. In other words, if the original function had an output value of y at a particular input value x, the transformed function will have an output value of y – 10 at the same input value x. Geometrically, this means that every point on the graph of f(x) is shifted downward by 10 units. To see why, imagine taking a point (x, y) on the original graph of f(x). When we apply the transformation f(x)↦f(x)–10, the new output value is y – 10. This means that the point (x, y – 10) is on the graph of the transformed function. Since y – 10 is less than y for every point on the original graph, this new point is lower down on the coordinate plane. Therefore, the entire graph of the transformed function is shifted downward by 10 units. In summary, the transformation f(x)↦f(x)–10 shifts the graph of f(x) downward by 10 units.