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 and the transformation applied. The base function here is \(y = x^{2}\), which is a standard parabola centered at the origin.
2. Recognize that the function \(ƒ(x) = x^{2} - 3\) is a vertical shift of the base function \(y = x^{2}\).
3. Understand that subtracting a constant from the function, as in \(x^{2} - 3\), shifts the graph vertically downward by that constant value.
4. Therefore, the graph of \(ƒ(x) = x^{2} - 3\) is the graph of \(y = x^{2}\) shifted down by 3 units.
5. Fill in the blank with the number 3, indicating the downward shift in units.

Key Concepts

Graphing Quadratic Functions

Graphing quadratic functions involves plotting parabolas based on the equation y = ax² + bx + c. The basic shape is determined by the coefficient a, while the position is influenced by b and c. Understanding how changes in the equation affect the graph is essential for accurate plotting.

Vertical Shifts of Graphs

A vertical shift moves the entire graph up or down without changing its shape. Adding or subtracting a constant k to the function, as in y = f(x) + k, shifts the graph vertically by k units. Positive k shifts the graph up, while negative k shifts it down.

Interpreting Function Transformations

Function transformations describe how changes to the equation affect the graph's position and shape. Recognizing these transformations, such as shifts, stretches, and reflections, helps in quickly sketching or understanding the graph of modified functions.