Polynomial and Relative Functions Quiz

Relative Function

• A relative function, typically expressed as f(x), is a function derived from a known function, g(x), through transformations like shifts, stretches, reflections, or combinations.
• These transformations alter the original function's graph's position and shape.
• Horizontal shifts involve adding or subtracting a constant from the input (x), while vertical shifts involve adding or subtracting a constant to the output (f(x)).
• Stretching or compressing a function occurs when multiplying the input or output by a constant, changing the graph's width.
• Reflecting a function across an axis is achieved by multiplying the input or output by -1.

Polynomial Function

• A polynomial function is a function expressed as a sum of terms, where each term consists of a variable raised to a non-negative integer power multiplied by a constant coefficient.
• The general form is P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, where 'n' is a non-negative integer and 'a' coefficients are constants.
• The degree of a polynomial is the highest power of the variable present.
• Polynomial functions have smooth, continuous graphs with no breaks or sharp turns.
• End behavior is determined by the leading term, extending towards positive or negative infinity at large x values.
• Polynomial functions are fundamental in algebra with various applications.

Exponential and Logarithms

• Exponential functions are in the form f(x) = a^x, where 'a' is a positive constant (the base) and x is the exponent.

• The base 'a' is typically greater than 1.

• Exponential growth occurs when a > 1.

• Exponential decay happens when 0 < a < 1.

• The graph always passes through the point (0, 1).

• The exponential function is always positive.

• Logarithmic functions, the inverse of exponential functions, are defined as y = log_a(x), where a > 0 and a ≠ 1.

• Logarithms solve equations of the form a^x = b.

• Base 10 logarithms are written as log₁₀(x), and natural logarithms use base 'e' (approximately 2.718), written as ln(x).

• Logarithms of products, quotients, and powers can be simplified.

• Logarithmic functions, used in science and engineering (e.g., pH), analyze growth/decay.

• The domain of logarithmic functions is (0, ∞) as negative or zero inputs are not possible.

Test your understanding of polynomial and relative functions with this quiz. Explore concepts such as horizontal and vertical shifts, stretching, and the characteristics of polynomial expressions. This challenge will reinforce your knowledge of function transformations and applications.