Work each problem. Find a function g(x)=ax+b whose graph can be obtained by translating the graph of ƒ(x)=2x+5 up 2 units and to the left 3 units.

Key Concepts

Function Translation

Function translation involves shifting the graph of a function without altering its shape. A vertical translation moves the graph up or down, while a horizontal translation shifts it left or right. For example, translating a function f(x) up by k units adds k to the function, resulting in f(x) + k.

Linear Functions

Linear functions are mathematical expressions of the form g(x) = ax + b, where 'a' represents the slope and 'b' the y-intercept. The slope indicates the steepness of the line, while the y-intercept is the point where the line crosses the y-axis. Understanding the characteristics of linear functions is essential for manipulating and translating them.

Combining Translations

When translating a function, multiple transformations can be combined. For instance, to translate f(x) = 2x + 5 up 2 units, you add 2 to the function, resulting in f(x) + 2. To shift it left by 3 units, you replace x with (x + 3). The final function incorporates both transformations, demonstrating how to effectively combine shifts.