Problem 15 - 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 \(x_{2}=5\)

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x) = x^{2} + 2x\) from \(x_{1} = 3\) to \(x_{2} = 5\) is 10.

Step 1: Substitute into the function

Firstly, substitute \(x_{1}\) and \(x_{2}\) into the function \(f(x) = x^{2} + 2x\). So, we get \(f(x_{1}) = f(3) = 3^2 + 2(3) = 9 + 6 = 15\) and \(f(x_{2}) = f(5) = 5^2 + 2(5) = 25 + 10 = 35\).

Step 2: Compute the average rate of change

Now that we have \(f(x_{1})\) and \(f(x_{2})\), we can compute the average rate of change using the formula \((f(x_{2}) - f(x_{1})) / (x_{2} - x_{1})\). Substituting into the formula, we get \((35 - 15) / (5 - 3) = 20 / 2 = 10\).

Step 3: Interpret the result

The average rate of change of the function \(f(x) = x^2 + 2x\) from \(x_{1} = 3\) to \(x_{2} = 5\) is 10. Which means the function increases by an average rate of 10 for every unit increase in \(x\) within the interval [3, 5].

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function characterized by the highest degree of its variable being two. In mathematical terms, it's represented as \(f(x) = ax^2 + bx + c\). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \(a\). In our exercise, the function \(f(x) = x^2 + 2x\) is quadratic because its highest term is \(x^2\).
Here:

• The coefficient \(a\) is 1, indicating the parabola opens upwards.
• The coefficient \(b\) is 2, which affects the tilt of the parabola.

Understanding the nature of quadratic functions can help you predict their behavior over various intervals.

Function Evaluation

Function evaluation is the process of finding the output, or \(y\)-value, of a function for a given \(x\)-value by substituting this \(x\) into the function. In this exercise, we evaluated the quadratic function \(f(x) = x^2 + 2x\) at two different points, \(x_1 = 3\) and \(x_2 = 5\).
To perform this step, substitute:

• \(x = 3\) into \(f(x)\) to get \(f(3) = 3^2 + 2(3) = 9 + 6 = 15\)
• \(x = 5\) into \(f(x)\) to get \(f(5) = 5^2 + 2(5) = 25 + 10 = 35\)

This process allows us to find the specific values needed for calculating the average rate of change over a specified interval.

Intervals

An interval refers to the range of \(x\)-values over which the behavior of a function is analyzed. In this exercise, the interval is from \(x_1 = 3\) to \(x_2 = 5\). Intervals can be open (not including endpoints) or closed (including endpoints), and may represent finite or infinite ranges.
For this function:\[ x \in [3, 5] \]
This closed interval includes both endpoints 3 and 5. Understanding intervals helps in analyzing how a function behaves within specific ranges, as well as in calculating averages such as the average rate of change.

Algebraic Expressions

Algebraic expressions consist of constants, variables, and arithmetic operations such as addition and multiplication. They can include simple expressions like \(x + 2\), or more complex ones like \(x^2 + 2x\), as in our exercise.
Key components in algebraic expressions include:

• Variables: Symbols that represent numbers, such as \(x\).
• Coefficients: Numbers multiplying the variables, like 2 in \(2x\).
• Constants: Standalone numbers, although there is no constant in our equation.

Learning to manipulate algebraic expressions is crucial for evaluating functions, solving equations, and understanding mathematical relationships.