In Exercises 33-44, use the graph of y = f(x) to graph each function g. g(x) = f(x+2)

Verified step by step guidance

1. Identify the transformation applied to the function $f(x)$. The function $g(x)=f(x+2)$ represents a horizontal shift.
2. Understand that the expression $f(x+2)$ indicates a shift to the left by 2 units. This is because adding a positive number inside the function's argument moves the graph in the negative direction along the x-axis.
3. Take the graph of $y=f(x)$ and move every point on this graph 2 units to the left. For example, if a point on $f(x)$ is at (3, 5), it will move to (1, 5) on $g(x)$.
4. Ensure that the shape and orientation of the graph remain unchanged. Only the position of the graph is altered by the horizontal shift.
5. Sketch the new graph of $g(x)=f(x+2)$ using the shifted points to accurately represent the transformation.

Key Concepts

Function Transformation

Function transformation refers to the process of altering the graph of a function through shifts, stretches, or reflections. In this case, the function g(x) = f(x + 2) represents a horizontal shift of the graph of f(x) to the left by 2 units. Understanding how transformations affect the position and shape of a graph is crucial for accurately graphing the new function.

Horizontal Shifts

Horizontal shifts occur when a function is modified by adding or subtracting a value from the input variable. For g(x) = f(x + 2), the '+2' indicates that every point on the graph of f(x) moves 2 units to the left. This concept is essential for predicting how the original graph will change and for accurately plotting the new function.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visually represent the relationship between the input (x) and output (y) values of a function. To graph g(x) = f(x + 2), one must first understand the original graph of f(x) and then apply the horizontal shift. This skill is fundamental in algebra as it helps in visualizing and interpreting mathematical relationships.