The x-axis is the horizontal axis on a coordinate plane, typically running left to right. It is used to represent the independent variable in a graph and helps visualize the relationship between two or more variables.
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The x-axis is the horizontal line that runs left to right on a coordinate plane, representing the independent variable.
On a graph, the x-axis is used to plot the independent variable, which is the variable that is manipulated or changed.
The domain of a function is the set of all possible input values, which are represented on the x-axis of a graph.
Transformations of functions, such as translations, reflections, and stretches, can be visualized by observing changes in the position and orientation of the graph on the x-axis.
When modeling linear functions, the x-axis represents the independent variable, which is the input value that is used to calculate the dependent variable, or the output value.
Review Questions
Explain the role of the x-axis in the rectangular coordinate system and how it is used to graph functions.
The x-axis is the horizontal axis in the rectangular coordinate system, which is used to represent the independent variable. When graphing functions, the x-axis is used to plot the input values, or the independent variable, while the y-axis is used to plot the output values, or the dependent variable. The relationship between the independent and dependent variables is then visualized on the coordinate plane, with the x-axis serving as the foundation for understanding the function's domain and behavior.
Describe how the x-axis is used to determine the domain and range of a function.
The domain of a function is the set of all possible input values, which are represented on the x-axis of a graph. The range of a function is the set of all possible output values, which are represented on the y-axis. By analyzing the x-axis, you can determine the minimum and maximum values of the domain, as well as any restrictions or limitations on the input values. This information is crucial for understanding the function's behavior and its potential applications.
Analyze how transformations of functions, such as translations, reflections, and stretches, are reflected in the position and orientation of the graph on the x-axis.
$$\text{When a function is transformed, the changes are often most easily observed on the x-axis. For example:} \begin{align*} \text{Translations:} &\quad f(x) \to f(x \pm h) \quad \text{shifts the graph left/right on the x-axis} \\ \text{Reflections:} &\quad f(x) \to f(-x) \quad \text{reflects the graph across the y-axis} \\ \text{Stretches:} &\quad f(x) \to a f(x) \quad \text{stretches/compresses the graph horizontally on the x-axis} \end{align*} $$ Understanding how these transformations affect the position and orientation of the graph on the x-axis is crucial for visualizing and analyzing the behavior of transformed functions.