Graph of a function

Definition

The graph of a function is a visual representation of the relationship between the input values (or independent variable) and the output values (or dependent variable) of that function. Each point on the graph corresponds to an ordered pair, where the x-coordinate represents an input value and the y-coordinate represents the resulting output value. This visual representation allows us to easily analyze and interpret the behavior of functions, including their trends, intercepts, and continuity.

5 Must Know Facts For Your Next Test

1. The graph can take many forms depending on the type of function, including linear, quadratic, exponential, and more complex shapes.
2. Graphs can be plotted on a coordinate plane, which consists of two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis).
3. Key features of a graph include intercepts, where it crosses the axes, as well as maximums and minimums that indicate peaks and valleys.
4. A function's graph must pass the vertical line test; if a vertical line intersects the graph at more than one point, it is not a function.
5. Transformations such as translations, reflections, and stretches can alter the appearance of a function's graph without changing its essential characteristics.

Review Questions

• How does the graph of a function visually represent its relationship between input and output values?
  • The graph of a function visually represents the relationship between input values and output values by plotting ordered pairs on a coordinate plane. Each point on the graph corresponds to an input-output pair, with the x-coordinate representing the input value and the y-coordinate representing the output value. This visual representation allows us to observe trends in how changes in input affect output, making it easier to analyze functions at a glance.
• What are some key features to look for when analyzing the graph of a function?
  • When analyzing the graph of a function, important features to look for include intercepts (where the graph crosses the axes), maximums and minimums (peaks and valleys), as well as overall trends like increasing or decreasing sections. Additionally, understanding whether the function is continuous or has breaks can provide insight into its behavior. These features help in predicting outputs for given inputs and understanding how the function behaves overall.
• Evaluate how transformations affect the appearance and properties of a function's graph while preserving its core characteristics.
  • Transformations such as translations, reflections, and stretches modify the appearance of a function's graph but maintain its fundamental characteristics. For example, translating a graph up or down shifts all points vertically without altering their relationships. Similarly, reflecting across an axis changes how the graph looks while keeping its points' relative positions intact. Stretches can make it wider or narrower while keeping its shape. These transformations help illustrate how functions can be manipulated while still reflecting similar behaviors in terms of domain and range.

Related terms

Ordered Pair: A pair of numbers used to represent a point on a graph, typically written in the form (x, y), where x is the input value and y is the output value.

Domain: The set of all possible input values for a function, which defines the range of x-coordinates on the graph.

Range: The set of all possible output values for a function, corresponding to the range of y-coordinates on the graph.