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3.2 Families of Graphs

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Family of graphs – a group of graphs that displays one or more similar characteristics Parent graph – basic graph that is transformed to create other members in a family of graphs. Parent graph – basic graph that is transformed to create other members in a family of graphs. Reflections and translations of the parent function can affect the appearance of the graph. The transformed graph may appear in a different location but it will resemble the parent graph. Reflections and translations of the parent function can affect the appearance of the graph. The transformed graph may appear in a different location but it will resemble the parent graph.

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Reflection – flips a figure over a line called the axis of symmetry. y = -f(x) is reflected over the x-axis. y = f(-x) is reflected over the y-axis.

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Ex 1 Graph f(x) = x 3 and g(x) = -x 3. Describe how the graphs are related.

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Translations – when a constant c is added or subtracted from a parent function, the result f(x) + or – c, is a translation of the graph up or down. When a constant c is added or subtracted from x before evaluating a parent function, the result f(x + or – c) is a translation left or right. When a constant c is added or subtracted from x before evaluating a parent function, the result f(x + or – c) is a translation left or right. y = f(x) + c: up c y = f(x) + c: up c y = f(x) – c: down c y = f(x) – c: down c y = f(x + c): left c y = f(x + c): left c y = f(x – c): right c y = f(x – c): right c

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Ex 3 Use the parent graph y = x 3 to graph the following: y = x 3 – 1 y = (x – 1) 3 y = (x – 1) 3 + 3

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Dilation – shrinking or enlarging a figure Dilation – shrinking or enlarging a figure When the leading coefficient of x is not 1, the function is expanded or compressed When the leading coefficient of x is not 1, the function is expanded or compressed y = c(f(x)), c >1: expands vertically y = c(f(x)), c >1: expands vertically y = c(f(x)), 0

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Ex 4 Describe how the graphs are related.

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