Problem 61 - College Algebra Essentials

Question

find and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h}, h \neq 0 $$ for the given function. $$ f(x)=x^{2}-4 x+3 $$

Step-by-Step Solution

Answer

The simplified difference quotient for the function \(f(x) = x^{2} - 4 * x + 3\) is \(2 * x + h - 4\).

Step 1: Substitute \(x\) and \(x + h\) into the function

Firstly, substitute \(x + h\) into the function for the term \(f(x + h)\), and substitute \(x\) for the term \(f(x)\). This is done as follows: \\[\frac{( (x + h)^{2} - 4 * (x + h) + 3) - (x^{2} - 4 * x + 3)}{h} \]

Step 2: Simplify the equation

Expand the terms in the numerator and then simplify by canceling the common terms. Note that the \(h\) in the denominator doesn’t cancel out with any of the \(h\)'s in the numerator. \\[\frac{(x^2 + 2*x*h + h^2 - 4*x -4*h +3) - (x^2 - 4*x + 3)}{h} \]Which simplifies to \\[= \frac{2*x*h + h^2 - 4*h}{h}\]

Step 3: Cancel h from all the terms in the numerator

In this step, cancel \(h\) from all the terms in the numerator. The simplified difference quotient is the result. Note that \(h\) can be canceled from the numerator because it isn’t equal to zero (as mentioned in the problem). \\[= 2*x + h - 4 \]

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill that helps in making complex equations more manageable. When dealing with expressions like the difference quotient, it's all about breaking them down into simpler forms.

Here’s what you need to consider when simplifying:

• Start by expanding any expressions within the parentheses. This involves using distributive properties such as multiplying out \(x+h\) into \(x^2 + 2xh + h^2\).
• Identify and group similar terms together. This step makes it easier to see which terms can be combined or canceled out.
• Look for any common factors that can be factored out or canceled. In this exercise, the variable \(h\) is crucial as it appears in all terms of the simplified numerator.

The main goal is to reduce the expression to its simplest form, where each component is easily understood. Simplifying properly not only makes calculations easier but also helps in understanding the function's behavior.

Polynomial Functions

Polynomial functions are expressions that involve only the operations of addition, subtraction, multiplication, and whole number exponents on variables. In our difference quotient exercise, \(f(x) = x^2 - 4x + 3\) is a quadratic polynomial.

Some key characteristics of polynomial functions include:

• They have terms that are combined through addition or subtraction, where each term consists of a variable raised to a power and multiplied by a coefficient.
• The degree of a polynomial is determined by its highest exponent, which in this case is 2, making it a quadratic.
• Polynomial functions are essential for modeling real-world scenarios, where they often describe smooth and continuous phenomena.

Understanding how to work with polynomials is pivotal for calculus as they often form the basis for more complex functions and derivatives.

Calculus Preparation

Preparing for calculus involves grasping fundamental notions such as limits, derivatives, and function behavior. The difference quotient is a stepping stone to understanding derivatives, which measure how functions change.

What you should focus on while preparing for calculus:

• Begin by understanding how change is calculated. The difference quotient represents average rate of change over an interval \(h\)
• Become comfortable with algebraic manipulations, as seen with simplifying expressions and working with polynomials.
• Practice recognizing patterns and limits. In calculus, limits help determine the behavior of functions at points of interest, like points where functions have vertical tangents or discontinuities.

A solid foundation in these preparatory topics will make the transition to calculus smoother, facilitating deeper insights into limits, derivatives, and integrals.