Parent Functions and Their Characteristics
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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 y-
intercept (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) = ax
2
+ bx + c,
a, b and c are constants.
All even functions of x
n
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) = ax
3
+ bx
2
+ 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 x
n
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
Exponential
In an
exponential function
, the
independent variable is
in
the exponent,
thus exponential functions grow very
rapidly compared to polynomial
functions.
LOGARITHMIC
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 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.
This informative content explores different parent functions such as linear, quadratic, cubic, square root, reciprocal, exponential, and logarithmic functions. It delves into their unique properties, graphs, and functions, offering insights into domain, range, and key characteristics of each function type.
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Presentation Transcript
PARENT FUNCTIONS Linear Quadratic Cubic Square Root Reciprocal Exponential Logarithmic Absolute Value
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 y- intercept (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 xnresemble 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 xnresemble 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 Exponential ? ? = ?? In an exponential function, the independent variable is in the exponent, thus exponential functions grow very rapidly compared to polynomial functions.
LOGARITHMIC Logarithmic ? ? = log(?) 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 values carry the magnitude of the value, regardless of whether the value is positive or negative. This means that absolute values are always positive. Absolute Value: ? ? = ? They can usually be characterized in a graph by the sharp corner in the function.
