Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=3√x-2

Verified step by step guidance

1. Identify the basic function: The function \( f(x) = 3\sqrt{x-2} \) is a transformation of the square root function \( \sqrt{x} \).
2. Determine the transformation: The expression \( x-2 \) inside the square root indicates a horizontal shift. Specifically, the graph of \( \sqrt{x} \) is shifted 2 units to the right.
3. Identify the vertical stretch: The coefficient 3 in front of the square root indicates a vertical stretch by a factor of 3. This means that each y-value of the basic square root function is multiplied by 3.
4. Plot key points: Start by plotting the point where the function begins, which is at \( x = 2 \) (since \( \sqrt{x-2} \) is undefined for \( x < 2 \)). Calculate a few more points by substituting values of \( x \) greater than 2 into the function.
5. Sketch the graph: Use the key points and the transformations identified to sketch the graph. The graph should start at the point (2, 0) and rise to the right, reflecting the vertical stretch and horizontal shift.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visually represent the relationship between the input (x) and output (f(x)). Understanding how to identify key features such as intercepts, asymptotes, and behavior at infinity is essential for accurately depicting the function's shape.

Cube Root Function

The cube root function, represented as f(x) = 3√x, is a type of radical function that returns the number which, when cubed, gives the input x. This function is defined for all real numbers and has a characteristic shape that increases gradually, crossing the origin and having a domain and range of all real numbers.

Transformations of Functions

Transformations of functions involve shifting, stretching, or reflecting the graph of a function. In the given function f(x) = 3√x - 2, the '-2' indicates a vertical shift downward by 2 units, which alters the graph's position without changing its shape. Understanding these transformations is crucial for accurately graphing modified functions.