Shifting a graph Use a shift to explain how the graph of $\text{ƒ}\left(x\right)=\sqrt{-x^2+8x+9}$ is obtained from the graph of $g\left(x\right)=\sqrt{25-x^2}$ . Sketch a graph of ƒ.

Key Concepts

Graph Shifting

Graph shifting refers to the process of moving a function's graph horizontally or vertically without altering its shape. This is achieved by adding or subtracting constants to the function's input (x) or output (y). For example, if a function f(x) is shifted to the right by 3 units, the new function becomes f(x - 3). Understanding this concept is crucial for analyzing how changes in the function's equation affect its graphical representation.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. In the given question, the function f(x) = √(-x^2 + 8x + 9) can be analyzed by rewriting the expression under the square root as a quadratic, which helps in understanding its vertex and intercepts.

Square Root Functions

Square root functions are defined as f(x) = √(g(x)), where g(x) is a non-negative function. The graph of a square root function typically starts at a point on the x-axis and increases, reflecting the non-negative outputs of the square root. In the context of the question, understanding how the square root affects the shape and domain of the function is essential for sketching the graph of f(x) based on the transformations applied to g(x).