Problem 16 - College Algebra Essentials

Question

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). \(f(x)=x^{2}-2 x\) from \(x_{1}=3\) to \(x_{2}=6\)

Step-by-Step Solution

Answer

The average rate of change is 7.

Step 1: Evaluate the Function at Given Points

Substitute the given point values into the function to find the function values: \(f(x_1) = f(3) = 3^{2}-2(3) = 9-6 = 3\) and\(f(x_2) = f(6) = 6^{2}-2(6) = 36-12 = 24\)

Step 2: Apply the Formula

The formula for the average rate of change is: \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). Substituting the previously calculated function values and the given x values into the formula, we get: \(\frac{f(6)-f(3)}{6 - 3} = \frac{24 - 3}{6 - 3} = \frac{21}{3} = 7\)

Step 3: Conclusion

The average rate of change of the function \(f(x) = x^2 - 2x\) from \(x_{1}=3\) to \(x_{2}=6\) is 7.

Key Concepts

Quadratic Functions

Quadratic functions form the foundation of many mathematical concepts and have a crucial role in understanding polynomial functions. The standard form of a quadratic function is expressed as \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, with \(a\) not being zero. This function produces a parabola when graphed, which can open upwards or downwards depending on the sign of \(a\).

A key characteristic of quadratic functions is their vertex, the highest or lowest point on the graph. This point is significant when analyzing the function's behavior. Understanding quadratic functions helps in predicting and interpreting the function's outputs based on inputs, which is essential in various applications from physics to finance.

Function Evaluation

Function evaluation is the process of finding the value of a function given a particular input or set of inputs. This process is fundamental in mathematics as it helps to determine how a function behaves under certain conditions.

To evaluate a function, you substitute the input values into the given equation. For example, in the original exercise, the function \(f(x) = x^2 - 2x\) is evaluated at \(x = 3\) and \(x = 6\).

Here's how it works:

• Calculate \(f(3)\) by substituting 3 into the equation: \(3^2 - 2 \times 3 = 9 - 6 = 3\).
• Calculate \(f(6)\) similarly: \(6^2 - 2 \times 6 = 36 - 12 = 24\).

Function evaluation is a step-by-step process, and accuracy in these steps is vital for reaching the correct results.

Mathematical Formulas

Mathematical formulas are the backbone of solving problems in calculus and algebra, as they provide quick solutions to complex calculations. In this particular problem, we are dealing with the formula for the average rate of change.

The average rate of change formula is:

• \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\)

This formula calculates the slope of the line connecting two points on a graph of a function. By substituting the values obtained from function evaluation, you can find the rate at which the function changes over a specified interval. Understanding how and when to apply these formulas helps significantly in solving mathematical problems efficiently.

Rate of Change

The rate of change is an important concept that describes how one quantity changes in relation to another. It's a crucial concept in calculus and physics because it gives you insight into the speed or intensity of changes over an interval.

In the context of the problem, the average rate of change is calculated for the function \(f(x) = x^2 - 2x\) from \(x = 3\) to \(x = 6\).

The steps to finding the average rate of change include:

• Evaluating the function at the given points to find \(f(3)\) and \(f(6)\).
• Applying the average rate of change formula \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\).
• In this case, you discover the average rate of change is \(7\).

This value means that as \(x\) changes from 3 to 6, the function \(f(x)\) increases by an average of 7 units. Being able to calculate and understand the rate of change is pivotal for studying how systems evolve over time.