5.3 Linear vs Non-Linear Functions

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What is the definition of a linear function?

Back

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.

Front

What is the definition of a nonlinear function?

Back

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.

Front

How can you determine if a graph represents a function?

Back

A graph represents a function if every vertical line drawn through the graph intersects it at most once. This is known as the vertical line test.

Front

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 crosses the graph more than once, then the graph does not represent a function.

Front

What is the significance of the x-value in a function?

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In a function, each x-value must correspond to exactly one y-value. If an x-value corresponds to multiple y-values, the relation is not a function.

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What does it mean for a relation to be 'not a function'?

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A relation is 'not a function' if there exists at least one x-value that is associated with more than one y-value.

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What are some common forms of nonlinear functions?

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Common forms of nonlinear functions include quadratic functions (e.g., f(x) = ax^2 + bx + c), exponential functions (e.g., f(x) = a * b^x), and trigonometric functions (e.g., f(x) = sin(x)).

Front

What is the slope of a linear function?

Back

The slope of a linear function is a measure of the steepness of the line, represented by 'm' in the equation f(x) = mx + b. It indicates how much y changes for a unit change in x.

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What is the y-intercept of a linear function?

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The y-intercept of a linear function is the point where the graph intersects the y-axis, represented by 'b' in the equation f(x) = mx + b.

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How can you identify a linear function from a table of values?

Back

A function is linear if the differences between consecutive y-values are constant when the x-values increase by the same amount.