"Explain how you can tell the function f(x) = |x - 5| is not linear by using points on its graph. Make sure to include in your answer the Points you used to determine that the function is not linear."
Help please?
To determine whether a function is linear, we need to check if it follows the properties of linearity, which are defined as having a constant rate of change. Here's how you can analyze the function f(x) = |x - 5| to demonstrate that it is not linear using points on its graph:
1. Before we dive into specific points, let's briefly understand the equation |x - 5|. This equation represents the absolute value of the difference between x and 5. The absolute value ensures that the output is always positive or zero, regardless of whether the input is positive or negative.
2. Pick a few values of x and evaluate the corresponding outputs to plot points on the graph. Choose values that will help us observe the behavior of the function. Let's select x = 0, x = 5, and x = 10 as our points of interest.
For x = 0, f(0) = |0 - 5| = |-5| = 5.
For x = 5, f(5) = |5 - 5| = |0| = 0.
For x = 10, f(10) = |10 - 5| = |5| = 5.
3. Plot the points (0, 5), (5, 0), and (10, 5) on a Cartesian plane.
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0 5 10
4. By examining the graph formed by these three points, we can observe that the points do not fall on a straight line. Instead, they form a "V" shape.
5. In a linear function, regardless of the points selected, the resulting graph would always form a straight line. However, in this case, we see that the function f(x) = |x - 5| does not exhibit a constant rate of change since the graph changes direction at x = 5.
Therefore, by analyzing the points on the graph, we can conclude that the function f(x) = |x - 5| is not linear.