Subjects
Grades
The Properties of a Linear Function
Concept sheet | Mathematics
Secondary
3-4
Table of contents
In the following animation, experiment with the values of parameters |a| and |b| of the first-degree polynomial function (linear function) and observe the effect on the function’s properties.
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Properties |
Linear function of the form |y=ax+b| |
|---|---|
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Domain |
|\mathbb{R}| or depending on the context |
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Range (codomain) |
|\mathbb{R}| or depending on the context |
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|x|-intercept |
|\displaystyle x = \frac{-b}{a}| or replace |y| with |0|, and then isolate |x| |
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Sign of the function |
If |a>0| , the function is negative over |(-∞,\frac{-b}{a}]| and positive over |[\frac{-b}{a},+∞).| If |a<0| , the function is positive over |(-∞,\frac{-b}{a}]| and negative over |[\frac{-b}{a},+∞).| |
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|y|-intercept |
The value of |b|. |
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Extrema |
None, or depending on the context. |
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Increasing intervals |
If the slope is positive |(a>0),| the function is increasing over its entire domain. |
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Decreasing intervals |
If the slope is negative |(a<0),| the function is decreasing over its entire domain. |
Determine the various properties of the function |y = 2x + 1|.
It is very useful to draw a graph to help determine the properties.
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The domain of the function is the set of real numbers, denoted |\mathbb{R}|, since the function has no restrictions on its domain.
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The range of the function is the set of real numbers, denoted |\mathbb{R}|, since the function outputs every real number.
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The |x|-intercept of the function is calculated as follows: |\displaystyle x = \frac{-b}{a} = \frac{-1}{2}|. It is possible to replace |y| with |0|, and then isolate |x|.
||\begin{align} 0 &= 2x + 1 \\ -1 &= 2x \\ \displaystyle \frac{-1}{2} &= x \end{align}|| -
The sign of the function is negative over |(-∞,\frac{-1}{2}]| and positive over |[\frac{-1}{2},+∞).|
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The |y|-intercept is |b = 1.|
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The function has no extrema.
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The function is increasing, because |a>0.|