PARENT FUNCTIONS
PARENT FUNCTIONS
Linear
Quadratic
Cubic
Square Root
Reciprocal
Exponential
Logarithmic
Absolute Value
DOMAIN AND RANGE
LINEAR
Linear
The graph of a linear function is a line.
Lines are characterized by a slope,
usually denoted (m) and a position on
the plane, usually given by an x- or a yintercept (b).
Functional form (slope-intercept form):
f(x) = mx + b
Linear equations (standard form):
Ax + By = C
QUADRATIC
Quadratic
The graph of a quadratic function (and
only a quadratic function) is a parabola.
Parameters: vertex location, direction of
opening and steepness of rise or drop.
Function form: f(x) = ax2 + bx + c,
a, b and c are constants.
All even functions of xn resemble this
shape
CUBIC
Cubic
The graph of a cubic function is S-shaped or
"sigmoid." The parent function has inversion
symmetry about (x, y) = (0, 0). That point is
called an inflection point, the point where
the curvature of the graph changes sign.
Functional form: f(x) = ax3 + bx2 + cx + d,
where A, B, C and D are constants. Cubic
and higher functions are generally referred
to as polynomial functions.
All odd functions of xn resemble this shape.
SQUARE ROOT
Root
Root functions have an exponent of the
independent variable that is less than
one.
Root functions are characterized by
rapid growth for small values of x, with
slower growth as x increases.
Note that the domain (the "allowed"
values of x) of a root function with an
even exponent (like 1/2) is always [0,
∞), while an odd exponent (like 1/3)
gives (-∞, ∞).
RECIPROCAL
Reciprocal Function
A reciprocal function is one in which x is
in the denominator.
Because these functions have variables in
the denominator, the denominator can
approach zero for certain values of x,
which leads to asymptotic behavior. In
this example, both of the graph axes
are asymptotes
EXPONENTIAL
𝑓 𝑥 =𝑎
𝑥
Exponential
In an exponential function, the
independent variable is in the exponent,
thus exponential functions grow very
rapidly compared to polynomial
functions.
LOGARITHMIC
𝑓 𝑥 = log(𝑥)
Logarithmic
Logarithmic functions are the inverses of
exponential functions. They grow
continuously as the independent variable
grows, but the rate of growth diminishes,
too.
ABSOLUTE VALUE
Absolute Value: 𝑓 𝑥 = 𝑥
Absolute values carry the magnitude of
the value, regardless of whether the
value is positive or negative. This means
that absolute values are always positive.
They can usually be characterized in a
graph by the sharp corner in the
function.