Problem 78 - College Algebra with Modeling and Visualization

Question

Complete the following for the given \(f(x)\) (a) Find \(f(x+h)\) (b) Find the difference quotient of \(f\) and simplify. $$ f(x)=-4 x^{2}+1 $$

Step-by-Step Solution

Answer

(a) \( f(x+h) = -4x^2 - 8xh - 4h^2 + 1 \); (b) Difference quotient: \( -8x - 4h \).

Step 1: Substitute in the Function

To find \( f(x+h) \), substitute \( x+h \) into the function \( f(x) = -4x^2 + 1 \). This gives \( f(x+h) = -4(x+h)^2 + 1 \).

Step 2: Expand the Expression

Expand \( (x+h)^2 \) to get \( x^2 + 2xh + h^2 \). Substitute this into the expression for \( f(x+h) \): \( f(x+h) = -4(x^2 + 2xh + h^2) + 1 \).

Step 3: Simplify the Expression

Distribute the \(-4\) through the terms \( (x^2 + 2xh + h^2) \): \( f(x+h) = -4x^2 - 8xh - 4h^2 + 1 \).

Step 4: Compute the Difference Quotient

The difference quotient is given by \( \frac{f(x+h) - f(x)}{h} \). Compute \( f(x+h) - f(x) = (-4x^2 - 8xh - 4h^2 + 1) - (-4x^2 + 1) \).

Step 5: Simplify the Difference

Simplify \( f(x+h) - f(x) \) to \( -8xh - 4h^2 \).

Step 6: Divide by \( h \)

Divide by \( h \) to get \( \frac{-8xh - 4h^2}{h} = -8x - 4h \).

Step 7: Final Simplified Expression

The simplified expression for the difference quotient is \( -8x - 4h \).

Key Concepts

Function Notation

Function notation is a way of expressing the relationship between variables in mathematics. It is generally given as \( f(x) \), where \( f \) represents the function itself, and \( x \) represents the input value. When you substitute a number for \( x \), the function provides an output. Function notation makes it easy to understand different expressions of a function, like \( f(x) \), \( f(x+h) \), and helps in analyzing how they change when input values change. For instance, in the given exercise, \( f(x) \) is \(-4x^2 + 1\). By substituting \( x+h \) in place of \( x \), we can find \( f(x+h) \), providing insights into how the function behaves at a shifted input. Using function notation ensures consistency across calculations and algebraic manipulations, thus allowing for clear communication of mathematical concepts.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables raised to non-negative integer powers and coefficients. They can be as simple as a constant or involve multiple terms with different powers. In this exercise, the function \( f(x) = -4x^2 + 1 \) is a polynomial function of degree 2, specifically called a quadratic function. Polynomial functions are fundamental in mathematics due to their solvable equations and the smooth curves they produce on graphs. The degree of a polynomial gives us valuable information about its structure:

• It indicates the maximum number of solutions or roots the function can have.
• It tells us about the general shape of the graph, such as whether it opens upwards or downwards.

Understanding polynomial functions is crucial when working with algebraic expressions, especially in simplification and differentiation processes.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form, often to make calculations more manageable or to reveal essential properties of the function. In the difference quotient task from the exercise, simplification is a critical step. Firstly, we substitute \( x+h \) into \( f(x) \) and expand the square \( (x+h)^2 \). Then, by distributing and combining like terms, we simplify the expression to find the difference quotient. The initial complex expression \( -4(x^2 + 2xh + h^2) + 1\) reduces to \( -4x^2 - 8xh - 4h^2 + 1\) after distributing the \(-4\). This simplification helps us to compute the difference quotient, which involves subtracting \( f(x) \), and further reducing to \(-8x - 4h\). This process not only simplifies calculations but highlights the function's incremental rate of change, crucial for understanding its behavior. Mastering algebraic simplification is essential in handling more complex mathematical problems with confidence.