Answer to Question #134476 in Algebra for Joseph Se

Question

A. Give the definition of linear and quadratic function in terms of formula and geometric meaning.

B. Define exponential function and give its practical significance.

C. Define logarithmic function and give its practical significance.

Answer 1

a)

A linear function is a function that results to a graph of a straight line. It has dependent variable (y) and independent variable (x) which do not contain any power more than one. The general form of a linear function is $Ax + By + C= 0$ where $A, B,$ and $C$ are constants. This can also be in the form $y = f(x) = a + bx$

A quadratic function on the other hand, is a function that contain just one of the variables, but is raised to the second power. Quadratic functions when plotted, produce a parabola. Its general form is $y = ax^2 + bx + c$

b)

An exponential function is a function formed when an independent variable (x) is expressed as the exponent. It has a form of $f(x) = a^x$, where a is the base and x is the exponent. Its significant is that it can be used to describe a decay or growth. For instance, an exponential function such as $f(x) = ( \frac{1}{2})^x$ is an example of a decaying exponential, since it reduces as the value of $x$ increases.

c)

Logarithmic functions are simply the inverse of exponential functions. This is because any exponential functions can be expressed in logarithmic form. Likewise, every logarithmic function can be written in exponential form.

Logarithmic functions have the form $f(x) = y = log_b x$ which can be expressed in exponential form $x = b^y$ ( b remains the base).

Logarithmic functions are useful in that they help to work with very large numbers which, ordinarily are difficult to handle.