Fill in the blank(s) to correctly complete each sentence.
To graph the function ƒ(x) = x² - 3, shift the graph of y = x² down ___ units.

Verified step by step guidance

1. Identify the base function, which is \( y = x^2 \).
2. Recognize that the function \( f(x) = x^2 - 3 \) is a transformation of the base function.
3. Understand that subtracting a constant from the function \( y = x^2 \) results in a vertical shift downward.
4. Determine the number of units the graph is shifted by looking at the constant subtracted, which is 3.
5. Conclude that the graph of \( y = x^2 \) is shifted down 3 units to obtain \( f(x) = x^2 - 3 \).

Key Concepts

Graphing Quadratic Functions

Graphing quadratic functions involves plotting a parabola defined by the equation. The standard form of a quadratic function is ƒ(x) = ax² + bx + c, where 'a' determines the direction of the parabola (upward or downward), and 'c' represents the y-intercept. Understanding how changes in 'c' affect the graph's position is crucial for accurately shifting the graph.

Vertical Shifts

Vertical shifts in graphing occur when a constant is added to or subtracted from a function. For example, in the function ƒ(x) = x² - 3, the '-3' indicates a downward shift of the graph by 3 units. This concept is essential for understanding how the graph's position changes without altering its shape.

Parabola Characteristics

A parabola is a symmetric curve defined by a quadratic function, characterized by its vertex, axis of symmetry, and direction of opening. The vertex represents the minimum or maximum point of the parabola, depending on the sign of 'a'. Recognizing these features helps in visualizing how the graph transforms with vertical shifts.