Use the graph of y = f(x) to graph each function g. g(x) = f(x+2)

Verified step by step guidance

1. Understand the transformation: The function g(x) = f(x+2) represents a horizontal shift of the graph of f(x). Specifically, adding 2 inside the parentheses shifts the graph 2 units to the left.
2. Identify key points on the graph of y = f(x): Look at the graph of f(x) and note the coordinates of key points, such as intercepts, peaks, valleys, or other significant points.
3. Apply the horizontal shift: For each key point (x, y) on the graph of f(x), subtract 2 from the x-coordinate to find the corresponding point on the graph of g(x). The new point will be (x-2, y).
4. Plot the shifted points: Using the transformed points, plot the new graph of g(x). Ensure that the shape of the graph remains the same as f(x), but shifted 2 units to the left.
5. Verify the transformation: Double-check that all points and features of the graph of g(x) match the expected transformation of f(x) shifted 2 units to the left.

Key Concepts

Function Transformation

Function transformation refers to the changes made to the graph of a function based on modifications to its equation. In this case, g(x) = f(x + 2) represents a horizontal shift of the function f(x) to the left by 2 units. Understanding how transformations affect the graph is crucial for accurately sketching the new function.

Horizontal Shifts

Horizontal shifts occur when the input variable of a function is altered by adding or subtracting a constant. 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 graph will change without recalculating every point.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of a function to understand its behavior and characteristics. By examining the graph of y = f(x), one can identify key features such as intercepts, maxima, and minima, which will help in accurately plotting g(x) after applying the transformation.