Step 1: Understand the equation. The equation given is ƒ(x) = x. This is a linear equation, which means it will graph as a straight line. The slope of the line is 1 and the y-intercept is 0.
Step 2: Draw a set of axes. Label the x-axis and the y-axis. The x-axis represents the input values (x) and the y-axis represents the output values (ƒ(x)).
Step 3: Plot the y-intercept. The y-intercept is the point where the line crosses the y-axis. Since the y-intercept is 0, plot a point at (0,0).
Step 4: Use the slope to find another point. The slope is the ratio of the change in y to the change in x. Since the slope is 1, this means that for every 1 unit increase in x, y also increases by 1 unit. So, from the y-intercept (0,0), move 1 unit to the right and 1 unit up to plot the point (1,1).
Step 5: Draw the line. Once you have at least two points, you can draw a straight line through them. This line represents the graph of the equation ƒ(x) = x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Definition
A function is a relation that assigns exactly one output for each input from its domain. In the case of ƒ(x) = x, the function maps every real number to itself, illustrating a direct linear relationship. Understanding this concept is crucial for interpreting how inputs and outputs relate in a graph.
Graphing linear functions involves plotting points that satisfy the equation on a coordinate plane. For ƒ(x) = x, the graph is a straight line that passes through the origin (0,0) with a slope of 1. This means for every unit increase in x, y also increases by one unit, creating a 45-degree angle with the axes.
The coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane is represented by an ordered pair (x, y). Understanding how to navigate this plane is essential for accurately graphing functions and interpreting their behavior visually.