Chapter 6 And 7 Summary

In Chapter 6 and 7, students learn how to preform operations with rational exponents and with inverse, exponential, and logarithmic functions. Rational, or fractional, exponents are powers where a base of a is manipulated by nth roots. For example, when n is equal to 2 or 3, an equation is referred to as a square root or a cube root respectively. In a square root, the radical’s answer must evaluate to a when multiplied by itself. Similarly, in the root of a cube an answer multiplied by itself twice must equal a Rational exponents can be written in either exponential form and as a fraction as in (am/n) or in the radical form (n √am ). When a function is in radical form n is referred to as the root index. When solving equations with radical

If a radical equation is written as (y = n √am ), it is called a radical function and can be graphed. Although Chapter 6 and 7 do not illustrate how all radical functions could be graphed, they do educate students on graphing functions of square root and cube root radicals. In inverse functions, the composition of two functions is equal. When functions are composed they are set as each other’s x values; the functions [g(x)] and [f(x)] can be composed as either [g(f(x))] or [f(g(x))]. If the two functions are inverses then both [g(f(x))] and [f(g(x))] will simplify to x. When a function is exponential, it takes the form (y=abx), and varies on a curved slope at increasing ratios. On a graphing calculator, the equation for (y=abx) can be found from a table using exponential regression. When (a > 0) and (b>1) in an exponential function, the function is said to display exponential

In a graph, an exponential growth function decreases towards the asymptote. As x moves toward negative infinity, y moves towards infinity. To further their understanding of exponential function, students simplify and use natural base e, also known as Euler’s number, in their exponential graphs during Chapters 6 and 7. In addition to being used in exponential graphs, natural base e can be applied to logarithmic functions and logarithms. Logarithms are used to find what power (x) a base (b) must be raised towards to give a certain number (y) they are described as logby = x if and only if bx=y. When Euler’s number is equal to the base(b), a logarithm is called a natural log. When b is equal to 10, logarithms are called common logs. Chapters 6 and 7 teach that logarithms can often be simplified, or rewritten, using logarithm properties, including the product property, the quotient property and the power property. Logarithm properties can also be applied to solve exponential and logarithmic equations. Simplifying logarithms in logarithmic equations allows logs to either be divided out of the problem or converted into exponents to make solving a problem