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Linear Functions
1. Linear functions are functions of a straight line.
2. The slope-intercept form is the most-used equation form of representing a straight line.
The slope-intercept form has two of the most essential components necessary to form a
straight line equation (generally called a linear equation because the graph of a straight
line is called a linear plot)
Slope-Intercept form of an equation:
Y = mx + b ;
m represents the slope or steepness or tilt of the line (if m is positive, the line is tilted up,
if m is negative, the line is tilted down. If m is 0, the slope has no tilt; we have a flat
line and if we have a vertical line that lies from top to bottom, we have an undefined or
infinite slope)
b represents the starting y-value or the y-value when x = 0 (Y-intercept)
Y_intercept (b)
The b value is also referred to as the y-intercept, or the point on y where the graph
crosses the y-axis. It can be zero, but it doesn’t always have to be. When b = 0, the line
can be shown as y = mx; this special case of a linear function is called a direct variation
(DV). It present y as having a staring value of zero and this function is also represented
as y = kx where y varies directly with x; the k is defined as a constant. The slope is also a
constant value for a line and it is identical to k. k is written instead of m to avoid
confusion with including the b always with y = mx + b since b in the case of y = mx or y
= kx is always 0.
3. Linear Functions can also be written in the standard form.
4. The standard form of a linear function is Ax + By = C where the coefficient A of x is
never a negative number and C is a constant value.
Ax + By = C can be re-written as y = mx + b as such:
Ax + By = C
By = C – Ax
Y = (C-Ax) / (B) or Y = C/B – (Ax)/B
5. Another important feature of a linear equation is the x-intercepts or the x-values where
we have no y-values. This can be thought of having some input (x), but no outputs (y).
6. The graph of a linear function (straight line) includes both a domain (the set of all
possible x values and a range; the set of all y-values.
7. Linear functions can be transformed vertically along the y-axis and horizontally along the
x-axis, depending on if the input (x) or output (y or f(x) is affected.
8. Y-notation and f(x) also called a function notation both refer to the set of y-values that
can be manipulated very easily through transformations.
9. The most basic of linear functions is the function y or f(x) = x also called y = x+ 0; here
the slope (m) = 1 because we have 1X value only and the y-intercept or b is 0.
10. The quadratic equation is a product of two linear equations.
Quadratic Functions