For Exercises 7 through 22, graph the equations by plotting...

For Exercises 7 through 22, graph the equations by plotting points. Identify the graph as linear or nonlinear. For linear equations, graph at least three points. For more complicated equations, graph at least five points. Label and scale the axes.
$$
y=-2 x^{2}+15
$$

Key Concepts

Graph Scaling and Labeling

Proper graph scaling and labeling are critical for accurately representing a function. Scaling ensures that the plotted points are proportionate to the defined units, while axis labeling provides clear identification of the values represented on the graph. This improves readability and interpretation of the graph, particularly when comparing multiple functions or analyzing specific properties.

Quadratic Functions

Quadratic functions are a type of nonlinear function characterized by the presence of an x² term, resulting in a parabolic graph that opens either upward or downward. They have a vertex that represents the maximum or minimum value of the function and can be described by the standard form y = ax² + bx + c, where a, b, and c are constants.

Graphing Functions

Graphing functions involves plotting a set of points that satisfy the relationship defined by the equation and then drawing a curve or line that connects these points. This visual representation helps in understanding the behavior, trends, and characteristics of the function over its domain.

Plotting Points

Plotting points is the process of selecting specific input values (typically x-coordinates) and computing the corresponding output values (y-coordinates) from the function. These coordinate pairs are then placed on the graph to provide a structured visualization of the function, which is essential for constructing an accurate graph.

Linear and Nonlinear Functions

Understanding whether a function is linear or nonlinear is crucial in graphing. A linear function produces a straight line and is represented by the equation y = mx + b, where m and b are constants, while a nonlinear function produces a curve. This distinction helps determine the appropriate method and number of points needed to accurately depict the function's behavior.

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