Types of Functions Quiz

Types of Functions

• Functions are relationships between inputs (domain) and outputs (range) where each input corresponds to exactly one output.

Linear Functions

• A linear function has the form f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept.
• The graph of a linear function is a straight line.
• The slope represents the rate of change of the function. A positive slope indicates an increasing function, a negative slope indicates a decreasing function, and a zero slope indicates a constant function.

Quadratic Functions

• A quadratic function has the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not zero.
• The graph of a quadratic function is a parabola.
• The parabola opens upwards if 'a' is positive and downwards if 'a' is negative.
• The vertex of the parabola represents the maximum or minimum value of the function.

Polynomial Functions

• A polynomial function is a function that can be expressed as a sum of terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power.
• The general form is f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, where 'a'_i are constants, and 'n' is a non-negative integer (degree).
• Examples include linear, quadratic, and cubic functions.
• The degree of the polynomial determines the general shape of the graph.

Rational Functions

• A rational function is a function that can be expressed as the quotient of two polynomial functions.
• The general form is f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions, and q(x) is not equal to zero.
• The graph of a rational function may have vertical asymptotes (where the denominator is zero) and/or horizontal asymptotes (depending on the degrees of the polynomials).

Exponential Functions

• An exponential function has the form f(x) = a^x, where 'a' is a positive constant and 'a' ≠ 1.
• The variable 'x' is in the exponent.
• The graph of an exponential function either increases or decreases depending on whether 'a' is greater than or less than 1 but greater than 0.

Logarithmic Functions

• A logarithmic function is the inverse of an exponential function and has the form f(x) = log_a(x), where 'a' is a positive constant and 'a' ≠ 1.
• The base 'a' is usually 10 (common logarithm) or e (natural logarithm).

Piecewise Functions

• A piecewise function is defined by different rules for different intervals of the input variable (domain).
• The function is expressed as a set of different functions each applied on a specific interval.

Absolute Value Functions

• An absolute value function is a function that always outputs a non-negative value.
• The general form is f(x) = |x|, where for any input x, the output is the magnitude of x. (the magnitude or distance from zero)
• The graph of an absolute value function has a 'V' shape.

Trigonometric Functions

• Trigonometric functions relate angles of a right-angled triangle to ratios of its side lengths.
• Common trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent.
• These functions are periodic and are often used in modelling repetitive phenomena.

Other Important Concepts

• Domain and range
• Asymptotes
• Intercepts (x-intercepts, y-intercepts of a function).
• Increasing and decreasing intervals
• Continuity
• Even and odd functions and their properties.
• Identifying the appropriate function type for a given problem.
• Understanding different function behaviors (growth, decay, etc.)
• Graphing different function types correctly.
• Evaluating a function by substituting values into the function equation.
• Transforming functions (shifting, stretching, compressing).
• Describing transformations from an equation or graph.
• Use of function notation (f(x), g(x))
• Properties and characteristics of functions.
• Application of functions in different areas (e.g. modeling real-world phenomena)

Test your understanding of different types of functions, including linear, quadratic, and polynomial functions. This quiz covers essential concepts such as the general forms of each function type, their graphs, and characteristics. Perfect for students exploring algebra and functions.