Problem 51 - College Algebra with Modeling and Visualization

Question

(Refer to Example 3.) Use the given \(f(x)\) to complete the following. (a) Calculate the average rate of change of \(f\) from \(x=1\) to \(x=2\) (b) Illustrate your result from part (a) graphically. $$ f(x)=x^{2} $$

Step-by-Step Solution

Answer

The average rate of change of \(f(x) = x^2\) from \(x=1\) to \(x=2\) is 3.

Step 1: Understand the Concept of Average Rate of Change

The average rate of change of a function over an interval from \(x=a\) to \(x=b\) is given by the formula \(\frac{f(b) - f(a)}{b-a}\). This represents the slope of the secant line through the points \((a, f(a))\) and \((b, f(b))\) on the graph of the function.

Step 2: Calculate the Values of the Function at the Given Points

Let's calculate \(f(1)\) and \(f(2)\) for the function \(f(x) = x^2\).- \(f(1) = 1^2 = 1\)- \(f(2) = 2^2 = 4\)

Step 3: Apply the Average Rate of Change Formula

Now, use the formula for the average rate of change: \[\frac{f(2) - f(1)}{2-1} = \frac{4 - 1}{2 - 1} = \frac{3}{1} = 3\]

Step 4: Graphically Illustrate the Result

To graphically illustrate the average rate of change, plot the function \(f(x) = x^2\) on a graph. Mark the points \((1, 1)\) and \((2, 4)\). Draw a line (the secant line) through these points. The slope of this line represents the average rate of change from \(x=1\) to \(x=2\), which is 3. The straight line will visually show how the function changed between these two points.

Key Concepts

Secant Line

In understanding the concept of the average rate of change of a function, the idea of a secant line becomes vital. The secant line is essentially a straight line that connects two points on the curve of a function. For the given quadratic function, which is a parabola, the secant line links the points \(a, f(a)\) and \(b, f(b)\) on the graph.

What does this mean? Well, the slope of this secant line is precisely the average rate of change over the interval \[a, b\]. This slope gives us an idea about how much and in what direction the function is changing within that specific range.

• The steeper the secant line, the more rapid the change of the function over the interval.
• If the line rises from left to right, this indicates an increase in values.
• If it falls, it shows a decrease.

In the context of the exercise, the secant line between the points \(1, 1\) and \(2, 4\) shows an upward slope which confirms the positive rate of change calculated as 3.

Quadratic Functions

Quadratic functions, such as \(f(x) = x^2\), are fundamental in mathematics and have a distinctive "U"-shaped graph called a parabola. The equation \(f(x) = x^2\) is the simplest form of a quadratic function. It is symmetric around the y-axis, with its vertex at the origin \(0, 0\).

Key characteristics of quadratic functions:

• The degree of the polynomial is 2, making it a quadratic.
• The graph is a smooth curve with a single minimum or maximum point.
• The function has constant second-order differences when calculated over equal intervals.

In our exercise, understanding these properties helps us anticipate the general shape and behavior of the function \(f(x) = x^2\) over any interval, like from \(x=1\) to \(x=2\). Computing the values at these specific points ensures precise calculations of changes.

Graphical Representation

Visualizing mathematical concepts, like the average rate of change, can be greatly simplified through graphical representation. In this scenario, graphing the function \(f(x) = x^2\) offers a clear picture of the behavior of the function between specified points.

Here's how you can visualize this:

• Plot the quadratic function on a coordinate grid.
• Mark key points—like \(1,1\) and \(2,4\) from our example.
• Draw the secant line that passes through these two points.

By observing this graph:

• The curve representing \(f(x) = x^2\) should be apparent as a symmetrical parabola.
• The secant line's slope demonstrates the rate at which the function's value increases within the chosen interval.

Graphical methods not only enhance understanding but also allow for easier detection of trends and behaviors in functions that might be less obvious from purely algebraic solutions.