Question

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=100(1+\sqrt{x})^{4} $$

Short answer

Question: Determine the structure of the function \(y=100(1+\sqrt{x})^{4}\), given it is in the form \(y=k \cdot (h(x))^{p}\). Answer: The function is structured as follows: k = 100, p = 4, and h(x) = 1 + √x. The function can be written as \(y=100(1+\sqrt{x})^{4}\).

Step-by-step solution

Identify the parameters k and p

For the given function \(y=100(1+\sqrt{x})^{4}\), we can compare it with the general form \(y= k \cdot (h(x))^{p}\). We can see that the parameter k is 100, and the parameter p is 4.

Identify the function h(x)

Now that we have identified the parameters k and p, we can determine the function h(x). In the given equation, h(x) is inside the parenthesis that is being raised to the power of p. In this case, h(x) is equal to \(1+\sqrt{x}\).

Rewrite the function in the form \(y=k \cdot(h(x))^{p}\)

With all the parameters identified and the function h(x) determined, the final step is to rewrite the function in the required form: $$ y=k \cdot(h(x))^{p} $$ So our final answer should be: $$ y=100(1+\sqrt{x})^{4} $$ This matches the original equation, so we have successfully analyzed the function and determined its form.

Key concepts

Exponential Functions

Exponential functions are a type of mathematical expression where a constant base is raised to a variable exponent. They appear frequently in algebra and are essential in modeling growth processes like population growth or radioactive decay. These functions have the general form:

• The constant base is typically a fixed number.
• The exponent is a variable representing the independent variable, commonly denoted as "x".

When studying exponential functions, one can observe rapid increases or decreases depending on the base. If the base is greater than one, the function will describe growth; if it is between zero and one, it will describe decay. Understanding these functions helps you analyze situations in various fields like finance, biology, and physics.

Function Parameters

Function parameters are the vital components that define the specific behavior of a function. In an exponential function of the form \(y = k \cdot (h(x))^p\), the parameters \(k\), \(h(x)\), and \(p\) determine the shape and orientation of the function. - **\(k\) (Coefficient):** This parameter scales the function. If \(k\) is positive, the function maintains its original orientation. If negative, it flips.- **\(h(x)\):** A function or expression inside the base, which can transform the input \(x\). It might involve operations like addition or multiplication with fractions, decimals, or other roots.- **\(p\) (Exponent):** The power to which the function \(h(x)\) is raised. It dictates the steepness and direction of the curve. If \(p\) is large, the curve rises or falls steeply as \(x\) changes. Each parameter plays a crucial role, and together they construct the graph or solution set of the function, allowing us to visualize representations of data or relationships in algebra.

Exponentiation

Exponentiation is a mathematical operation involving a base and an exponent. It indicates that the base is raised to the power of the exponent, represented as \(a^n\). In this process, \(a\) is the base and \(n\) is the exponent, defining how many times to multiply the base by itself.Consider the following aspects of exponentiation:- **Basic Properties**: - For any number \(a\), \(a^1 = a\). - \(a^0 = 1\) for any non-zero \(a\). - \(a^n \times a^m = a^{n+m}\).- **Special Considerations**: - A negative exponent signifies reciprocal, for instance, \(a^{-n} = \frac{1}{a^n}\).- **Applications**: - Exponentiation is used to express large numbers compactly, as in scientific notation. - Powers of base ten are frequently used in fields dealing with very small or very large numbers, like physics and engineering. Mastering exponentiation aids in efficiently solving and manipulating expressions, particularly when dealing with exponential growth or decay equations in algebraic forms.