One example of an equation that is not a linear function is:
\[ y = x^2 + 3 \]
Explanation:
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Definition of Linear Function: A linear function is characterized by its graph being a straight line. In general, a linear function can be expressed in the form: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. The key characteristic of linear functions is that they have a constant rate of change.
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Quadratic Function: The equation \( y = x^2 + 3 \) is a quadratic function. It is a polynomial of degree 2 (the highest power of \( x \) is 2). The graph of this function is a parabola that opens upwards, which is distinctly different from a straight line.
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Non-Constant Rate of Change: The rate of change for \( y = x^2 + 3 \) is not constant. For example, if we take two points:
- When \( x = 1 \): \( y = 1^2 + 3 = 4 \)
- When \( x = 2 \): \( y = 2^2 + 3 = 7 \) The change in \( y \) (from 4 to 7) is 3, while the change in \( x \) (from 1 to 2) is 1, resulting in a slope of 3 between these two points.
- Now consider:
- When \( x = 2 \): \( y = 7 \) (from before)
- When \( x = 3 \): \( y = 3^2 + 3 = 12 \) The change in \( y \) (from 7 to 12) is 5, while the change in \( x \) (from 2 to 3) is still 1, resulting in a slope of 5 between these two points.
This demonstrates that the slope (rate of change) is not constant, verifying that \( y = x^2 + 3 \) is not a linear function.
Thus, \( y = x^2 + 3 \) is a valid example of an equation that is not a linear function because it does not satisfy the definition of a linear function, as evidenced by its non-linear graph and variable rate of change.