5.3 Linear vs Non-Linear Functions

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What is the definition of a linear function?

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A linear function is a function that can be graphically represented in the Cartesian plane as a straight line. It has the form f(x) = mx + b, where m is the slope and b is the y-intercept.

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What is a nonlinear function?

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A nonlinear function is a function that does not create a straight line when graphed. Its graph can be a curve, a circle, or any other shape that is not linear.

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How can you determine if a set of values represents a function?

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A set of values represents a function if each input (x-value) is associated with exactly one output (y-value). If any x-value is repeated with different y-values, it is not a function.

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What is the vertical line test?

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The vertical line test is a method used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.

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What does it mean for a relation to be a function?

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A relation is a function if every input is paired with exactly one output. This means no x-value can correspond to more than one y-value.

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What is an example of a linear function?

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An example of a linear function is f(x) = 2x + 3, which represents a straight line with a slope of 2 and a y-intercept of 3.

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What is an example of a nonlinear function?

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An example of a nonlinear function is f(x) = x^2, which represents a parabola and is not a straight line.

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What is the significance of the slope in a linear function?

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The slope in a linear function indicates the steepness of the line and the direction of the relationship between the variables. A positive slope means the line rises, while a negative slope means it falls.

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What is the y-intercept in a linear function?

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The y-intercept is the point where the line crosses the y-axis. It is represented by the value of b in the linear equation f(x) = mx + b.

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How can you identify a nonlinear relationship from a graph?

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A nonlinear relationship can be identified from a graph if the graph does not form a straight line, indicating that the rate of change between the variables is not constant.