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

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In this case, the function ƒ(x) = (x + 4)² represents a horizontal shift of the basic quadratic function y = x². Understanding how changes in the function's equation affect its graph is crucial for accurately completing the sentence.

Horizontal Shifts

A horizontal shift occurs when the graph of a function is moved left or right along the x-axis. For the function ƒ(x) = (x + 4)², the '+4' indicates a shift to the left by 4 units. This concept is essential for determining the correct direction of the shift when completing the sentence.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically represented in the form y = ax² + bx + c. The graph of a quadratic function is a parabola. Recognizing the standard form of a quadratic function helps in understanding how its graph behaves and how transformations affect its position on the coordinate plane.