key term - Constant Function

Definition

A constant function is a function that always returns the same output value, regardless of the input value. It is a special type of function where the output is fixed and does not depend on the input variable.

5 Must Know Facts For Your Next Test

1. The graph of a constant function is a horizontal line, as the output value remains the same regardless of the input value.
2. The rate of change (slope) of a constant function is always zero, as the output does not change with respect to the input.
3. Constant functions are often used to represent fixed quantities, such as a person's height or the speed of light.
4. In the context of power functions and polynomial functions, constant functions can be considered a special case where the exponent is zero.
5. Constant functions are important in calculus, as they have a constant derivative, which is always equal to zero.

Review Questions

• Explain how a constant function differs from a linear function in terms of its rate of change and graph.
  • The key difference between a constant function and a linear function is their rate of change. In a linear function, the output changes at a constant rate as the input changes, resulting in a straight line graph with a non-zero slope. In contrast, a constant function has a rate of change (slope) of zero, as the output value remains the same regardless of the input. This means the graph of a constant function is a horizontal line, rather than a line with a non-zero slope.
• Describe the role of constant functions within the context of power functions and polynomial functions.
  • Constant functions can be considered a special case of power functions and polynomial functions, where the exponent is zero. In a power function, $f(x) = x^n$, when $n = 0$, the function becomes a constant function, $f(x) = 1$, as any number raised to the power of zero is equal to one. Similarly, in a polynomial function, $f(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0$, the constant term $a_0$ represents a constant function, as it does not depend on the input variable $x$.
• Analyze the importance of constant functions in the context of rates of change and the behavior of graphs.
  • Constant functions play a crucial role in understanding rates of change and the behavior of graphs. Since the output of a constant function does not change with respect to the input, the rate of change (slope) is always zero. This means that constant functions have a flat, horizontal graph, which is an important concept in calculus when analyzing the behavior of functions and their derivatives. The constant nature of these functions also makes them useful for representing fixed quantities or parameters in various mathematical models and applications.

Related terms

Linear Function:

A linear function is a function where the output changes at a constant rate as the input changes. It has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Polynomial Function:

A polynomial function is a function that can be expressed as the sum of one or more monomials, where each monomial consists of variables raised to non-negative integer powers and multiplied by coefficients.

Exponential Function:

An exponential function is a function where the output grows or decays at a constant rate relative to the input, and can be expressed in the form $f(x) = a^x$, where $a$ is the base and $x$ is the input variable.