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12a: Increasing and Decreasing Functions © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

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Increasing and Decreasing Functions An increasing function is one whose gradient is always greater than or equal to zero. for all values of x A decreasing function has a gradient that is always negative or zero. for all values of x

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Increasing and Decreasing Functions e.g.1 Show that is an increasing function Solution: a positive number ( 3 ) a perfect square ( which is positive or zero for all values of x, and for all values of x is the sum of a positive number ( 4 ) so, is an increasing function

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Increasing and Decreasing Functions Solution: e.g.2 Show that is an increasing function. To show that is never negative ( in spite of the negative term ), we need to complete the square. is an increasing function. for all values of x Since a square is always greater than or equal to zero,

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Increasing and Decreasing Functions The graphs of the increasing functions and are and

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Increasing and Decreasing Functions Exercises 2. Show that is an increasing function and sketch its graph. 1. Show that is a decreasing function and sketch its graph. Solutions are on the next 2 slides.

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Increasing and Decreasing Functions 1. Show that is a decreasing function and sketch its graph. Solutions Solution:. This is the product of a square which is always and a negative number, so for all x. Hence is a decreasing function.

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Increasing and Decreasing Functions Solutions 2. Show that is an increasing function and sketch its graph. Solution:. Completing the square: which is the sum of a square which is and a positive number. Hence y is an increasing function.

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Increasing and Decreasing Functions

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