What is Affine function?

Also known as: slope · y-intercept

Definition 67.4 Primary & Middle School Mathematics · Chapter 67 — Linear and Affine Functions

An affine function has the form

f(x) = a x + b ,

where $a$ and $b$ are fixed numbers. Linear functions are the special case $b = 0$ ; constant functions the case $a = 0$ .

Examples

Example 67.8

Find the affine function with $f(2) = 7$ and $f(5) = 16$ .

a = \frac{16 - 7}{5 - 2} = \frac93 = 3 ,

then $f(2) = 3 \times 2 + b = 7$ gives $b = 1$ : $f(x) = 3x + 1$ . Check: $f(5) = 15 + 1 = 16$ .

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Definition 4.1 High School Mathematics · Chapter 4 — Reference Functions

An affine function is a function of the form

f(x) = mx + p,

where $m$ and $p$ are fixed real numbers. Its graph is a straight line: $m$ is the slope and $p$ the $y$ -intercept (the line crosses the vertical axis at $(0, p)$ ). When $p = 0$ the function is linear: $f(x) = mx$ expresses proportionality.

Two affine functions: increasing for m = 1/2 > 0, decreasing for m = -1 < 0. Moving one unit to the right changes y by m. — Two affine functions: increasing for $m = \frac12 > 0$ , decreasing for $m = -1 < 0$ . Moving one unit to the right changes $y$ by $m$ .

Examples

Example 4.3

Find the affine function whose graph passes through $A(1, 3)$ and $B(4, 9)$ . The slope is

m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2 .

Then $f(x) = 2x + p$ , and $f(1) = 3$ gives $2 + p = 3$ , so $p = 1$ : $f(x) = 2x + 1$ . Check with $B$ : $f(4) = 9$ .

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Examples

Examples

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