Algbra Unit 9

Linear Function

y = x, represented by a diagonal line that passes through the origin with a domain and range of all real numbers.

Quadratic Function

y = x², described by a U-shape with a vertex at (0, 0), with a domain of all real numbers and a range of [0, ∞).

Absolute Value Function

y = |x|, forms a V-shape with a vertex at (0, 0), having a range of [0, ∞) and a domain of all real numbers.

Square Root Function

y = √x, characterized by a curve starting at (0, 0) and moving right, with both domain and range of [0, ∞).

Inverse of a Linear Function

The inverse of a linear function is itself.

Exponential Growth Function

y = b^x (b > 1), presents a flat then fast increase, with a horizontal asymptote at y = 0, domain of all real numbers, and range of (0, ∞).

Rational Function

y = 1/x, consists of two branches with asymptotes at x = 0 and y = 0, with a domain and range of all reals except 0.

Horizontal Asymptote

A line that the graph approaches as x approaches ±∞, commonly found in exponential and rational functions.

Intercepts of a Quadratic Function

A quadratic function can have 0, 1, or 2 x-intercepts and always has 1 y-intercept.

Transformations - Vertical Shift

To shift a function vertically, use f(x) ± k.

Transformations - Reflection

Reflection over the x-axis is represented as -f(x) and over the y-axis as f(-x).

Extraneous Solutions

Solutions that arise from the algebraic manipulation of an equation but do not satisfy the original equation, common in rational, square root, and logarithmic functions.

Imaginary Solutions

Solutions that appear when the discriminant of a quadratic equation is negative, yielding no real number solutions.

Inverse of Exponential Function

The inverse of an exponential function is a logarithmic function.

Piecewise Function

A function defined by different rules for different parts of its domain.

Domain of Linear Function

The domain of a linear function is all real numbers.

Range of Quadratic Function

The range of a quadratic function is [0, ∞).

Inversion Relationship - Quadratic and Square Root

The quadratic function and square root function are inverses of each other if the quadratic is restricted to the non-negative range.

Logarithmic Function

y = log_b(x), is the inverse of the exponential function y = b^x, with a vertical asymptote at x = 0, domain of (0, ∞), and range of all real numbers.

Transformations - Horizontal Shift

To shift a function horizontally, use f(x ± h).

Transformations - Vertical Stretch/Compression

To stretch or compress a function vertically, use a * f(x).

Transformations - Horizontal Stretch/Compression

To stretch or compress a function horizontally, use f(bx).

Symmetry of Quadratic Function

Quadratic functions are symmetric about the vertical line passing through their vertex.

Range of Exponential Growth Function

The range of an exponential growth function is (0, ∞).

Domain of Square Root Function

The domain of a square root function is [0, ∞).

Critical Points

Points where the derivative of a function is either zero or undefined.

Vertex of a Parabola

The vertex of a parabola is the point where the parabola changes direction; it can be either a maximum or minimum point.

Axis of Symmetry

A vertical line that passes through the vertex of a parabola, dividing it into two equal halves.

Inversion Relationship - Exponential and Logarithmic

Logarithmic functions and exponential functions are inverses of each other.

Exponential Decay Function

A function that describes growth which decreases rapidly at first, then more slowly as time goes on.

Asymptote

A line that a curve approaches but never touches.

Maximum Point

The point on a function where the function reaches its highest value.

Non-permissible Values

Solutions that satisfy the simplified form of an equation but do not satisfy the original equation due to restrictions on the domain

Half-Open Interval

An interval that is defined by one endpoint that is inclusive and one that is exclusive.

Inflection Point

A point at which a function changes concavity.

Secant Line

The line that passes through two points on a curve.

Radical Function

A function that contains a radical expression with the independent variable in the radicand.

Range of a Function

The set of all possible output values of a function.

Non-permissible Values

Solutions that satisfy the simplified form of an equation but do not satisfy the original equation due to restrictions on the domain

Half-Open Interval

An interval that is defined by one endpoint that is inclusive and one that is exclusive.

Inflection Point

A point at which a function changes concavity.

Secant Line

The line that passes through two points on a curve.

A function that contains a radical expression with the independent variable in the