Fill in the blank(s) to correctly complete each sentence.
The graph of ƒ(x) = (x + 4)² is obtained by shifting the graph of y = x² to the ___ 4 units.

Verified step by step guidance

1. Identify the base function and the transformation function. The base function here is y = x^2, which is a standard parabola centered at the origin (0,0).
2. Analyze the transformation function ƒ(x) = (x + 4)^2. The transformation involves (x + 4) instead of x, indicating a horizontal shift.
3. Understand the direction of the shift. The +4 inside the parentheses with x indicates a shift to the left if it were -4, it would indicate a shift to the right.
4. Determine the magnitude of the shift. The number 4 represents the number of units the graph shifts from the original position of the base function.
5. Conclude the direction and magnitude of the shift. The graph of ƒ(x) = (x + 4)^2 is obtained by shifting the graph of y = x^2 to the left by 4 units.

Key Concepts

Function Transformations

Function transformations involve shifting, stretching, or reflecting the graph of a base function. In this case, adding a constant inside the function's argument shifts the graph horizontally. Understanding how these changes affect the graph is essential for interpreting and sketching functions.

Horizontal Shifts in Quadratic Functions

A horizontal shift occurs when a constant is added or subtracted inside the function's input, such as (x + h)². Specifically, f(x) = (x + 4)² shifts the graph of y = x² horizontally by 4 units. The sign inside the parentheses determines the direction of the shift.

Graph of the Basic Quadratic Function y = x²

The graph of y = x² is a parabola centered at the origin with its vertex at (0,0). It opens upwards and is symmetric about the y-axis. Recognizing this base graph helps in understanding how transformations like shifts affect its position.