Problem 24 - Algebra and Trigonometry

Question

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=4-x^{2} ; \quad x=1, x=1+h $$

Step-by-Step Solution

Answer

The average rate of change is \(-2-h\).

Step 1: Understand the Problem

We need to find the average rate of change of the function \( f(x) = 4 - x^2 \) as the variable \( x \) changes from \( x=1 \) to \( x=1+h \). The average rate of change is analogous to finding the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a graph of the function.

Step 2: Substitute Values into Function

Calculate \( f(x) \) at both points. Start by finding \( f(1) \) which is the value of the function when \( x=1 \). Substitute \( x=1 \) into the function: \[ f(1) = 4 - (1)^2 = 4 - 1 = 3 \] Now, calculate \( f(1+h) \), which is the value of the function when \( x = 1+h \). Substitute \( x=1+h \) into the function: \[ f(1+h) = 4 - (1+h)^2 = 4 - (1 + 2h + h^2) = 4 - 1 - 2h - h^2 = 3 - 2h - h^2 \]

Step 3: Calculate Average Rate of Change

The formula for the average rate of change is given as: \[ \text{Average Rate of Change} = \frac{f(1+h) - f(1)}{(1+h) - 1} \] Using the values calculated earlier: \[ f(1+h) - f(1) = (3 - 2h - h^2) - 3 = -2h - h^2 \] And since \((1+h) - 1 = h\), the average rate of change simplifies to: \[ \frac{-2h - h^2}{h} = -2 - h \]

Step 4: Express the Final Answer

The average rate of change of the function \( f(x) = 4-x^2 \) from \( x=1 \) to \( x=1+h \) is \(-2-h\). This expression shows how the rate of change varies depending on the incremental change \( h \).

Key Concepts

Functions

Functions are an essential concept in mathematics, especially when learning about changes and how variables relate to each other. A function, in simple terms, is a relationship between two variables where each input value (usually denoted as \( x \)) is related to exactly one output value (often represented as \( f(x) \)). This relationship can be visualized as a graph where each point connects the input to an output.

Understanding functions involves recognizing the inputs, the rule that defines the function, and the corresponding outputs. The function given in the original exercise is \( f(x) = 4 - x^2 \). This specific function is a quadratic function, but it serves as a classic example of how one can use functions to understand change.

Here is a simplified approach to understand functions:

• Identify the function rule: What equation or formula connects \( x \) to \( f(x) \)?
• Evaluate the function with specific input values to find output values. This is how we find points along the graph of the function.
• Graph the function to visually see how inputs relate to outputs.
• Use function values to determine rates of change or other properties, as demonstrated in the exercise.

Quadratic Functions

Quadratic functions are a special type of function that can be written in the general form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In the exercise, the function \( f(x) = 4 - x^2 \) is a quadratic function because it involves the square of the input variable \( x^2 \).

Quadratic functions have distinctive U-shape graphs known as parabolas, which can open upwards or downwards. The direction of the parabola depends on the coefficient \( a \). In this exercise, since the function is \( f(x) = -x^2 + 4 \), the negative sign before \( x^2 \) indicates that the parabola opens downward.

Key characteristics of quadratic functions:

• The vertex is the highest or lowest point on the graph. For a downward-opening parabola like \( f(x) = 4 - x^2 \), the vertex is the maximum point.
• The axis of symmetry is a vertical line passing through the vertex. This line helps divide the parabola into two mirror-image halves.
• The roots or zeros are the \( x \) values where the function equals zero, giving the points where the parabola intersects the \( x \)-axis.
• Quadratic functions are often analyzed to find their maximum or minimum values and to understand how they change as \( x \) changes.

Algebra

Algebra is the branch of mathematics dealing with symbols and rules for manipulating those symbols. It is like a language that allows us to solve equations and understand patterns. The exercise utilizes algebra to find the average rate of change of a quadratic function.

One of the primary skills in algebra is substituting variable values into equations to solve for unknowns. In the step-by-step solution, substitution helps evaluate \( f(x) \) at specific points, \( x = 1 \) and \( x = 1+h \).
Calculating the average rate of change is another application of algebra. This involves understanding and manipulating the formula:
\[\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}\]

• Identify two points \((a, f(a))\) and \((b, f(b))\) on the function graph.
• Substitute these points into the formula to calculate the rate of change. This helps determine how fast the function's output changes when the input changes.
• Simplify the expression as seen in the solution, where \(-2 - h\) represents how the rate of change varies with different \( h \) increments.

Using algebra in these ways shows how powerful it is for dissecting problems to find solutions, particularly in understanding and applying concepts like the average rate of change.