Problem 16 - College Algebra

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 of the function \(f(x) = x^{2} + 2x\) , from \(x_{1} = 3\) to \(x_{2} = 6\), is 11

Step 1: Substitute the Points into the Function

First, Evaluate function \(f(x)\) at two points: \(x_{1} = 3\) and \(x_{2} = 6\). Therefore you get \(f(x_{1}) = f(3) = (3)^{2} + 2(3) = 9 + 6 = 15\) and \(f(x_{2}) = f(6) = (6)^{2} + 2(6) = 36 + 12 = 48\).

Step 2: Calculate the Average Rate of Change

The average rate of change is given by the formula \(\frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}\). Substitute \(f(x_{1}) = 15\), \(f(x_{2}) = 48\), \(x_{1} = 3\) and \(x_{2} = 6\) into the formula to obtain \(\frac{48 - 15}{6 - 3} = \frac{33}{3} = 11\).

Step 3: Conclusion

Therefore, the average rate of change of the function \(f(x) = x^{2} + 2x\) from \(x_{1} = 3\) to \(x_{2} = 6\) is 11.

Key Concepts

Function Evaluation

Function evaluation is about finding the value of a function for a specific input. In mathematics, a function assigns exactly one output to each input value, much like a machine. To "evaluate" this means to compute or determine the function’s value at particular points.
Let's take an example. If we have a function defined as \(f(x) = x^2 + 2x\), when evaluating this function at \(x = 3\), we plug into the expression: compute \(3^2 + 2 \times 3 = 9 + 6 = 15\). Similarly, at \(x = 6\), it's computed as \(6^2 + 2 \times 6 = 36 + 12 = 48\).
Function evaluation is an essential step in many mathematical processes, including finding averages or rates of change.

Substitution

Substitution is a straightforward way to replace variables with specific values. It’s like filling in the blanks with given numbers.
In problems involving functions, substitution helps us find values by replacing the function’s variable with numbers provided in the question.
For example, in \(f(x) = x^2 + 2x\), when instructed to find \(f(3)\), we substitute \(x\) with 3: \((3)^2 + 2 \times 3 = 9 + 6 = 15\). Similarly, \(f(6)\) results in replacing \(x\) with 6, resulting in \(36 + 12 = 48\).
This technique is useful in breaking down complicated problems into simpler calculations.

Quadratic Function

A quadratic function is a type of function that can be expressed in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
The graph of a quadratic function is a parabola, which may open upwards or downwards depending on the sign of \(a\).

• When \(a > 0\): Parabola opens upwards.
• When \(a < 0\): Parabola opens downwards.

In our example, the function \(f(x) = x^2 + 2x\) is quadratic, with \(a = 1\), \(b = 2\), and \(c = 0\). The absence of a constant \(c\) means it passes through the origin when \(x = 0\).
Understanding the structure of quadratic functions is crucial for graphing and solving them.

Rate of Change Formula

The rate of change formula is essential in determining how one quantity changes relative to another. This concept is particularly useful in analyzing functions to see how their outputs change as their inputs shift.
The average rate of change between two points \(x_1\) and \(x_2\) on a function \(f(x)\) is calculated by: \[\frac{f(x_2) - f(x_1)}{x_2 - x_1}\]This formula gives the slope of the "secant line" connecting these points on the graph.
In our example, substituting \(f(6) = 48\) and \(f(3) = 15\), with \(x_2 = 6\) and \(x_1 = 3\), we find:\[\frac{48 - 15}{6 - 3} = \frac{33}{3} = 11\]This tells us the average change in value between these two points is 11 per unit and is crucial for understanding function behaviors over specific intervals.