To determine which situations represent linear functions, we need to assess whether the relationship between the two variables can be described with a linear equation (i.e., in the form \(y = mx + b\), where \(m\) is a constant rate of change).
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The temperature rises and falls over the course of a day. Temperature is a function of time.
- Not linear: The temperature typically varies in a non-linear manner (e.g., it might rise in the afternoon and drop at night).
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A baseball is hit into the outfield. The height of the ball is a function of time.
- Not linear: The height of the ball follows a parabolic trajectory, which means it is quadratic, not linear.
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A car goes 55 mph on the highway. Distance is a function of time.
- Linear: The distance traveled over time can be expressed as \(d = 55t\). This is a linear relationship.
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A child grew 2 inches this year and 1 inch last year. Height is a function of time.
- Not linear: The growth does not occur at a constant rate each year in this scenario, making it a variable change.
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A movie director makes 2 movies per year. The number of movies is a function of the years.
- Linear: The number of movies can be expressed as \(m = 2t\) (where \(t\) is the number of years). This is a linear relationship.
Based on this analysis, the situations that represent linear functions are:
- A car goes 55 mph on the highway. Distance is a function of time.
- A movie director makes 2 movies per year. The number of movies is a function of the years.
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