Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the function, approximate the relative minimum or maximum of the function, and estimate the open intervals on which the function is increasing or decreasing. \(f(x)=x^{2}-4 x+1\)

Step-by-Step Solution

Answer

The function \(f(x)=x^{2}-4x+1\) has a minimum at the point (2,-3), decreases on the interval \(- \infty, 2\) and increases on the interval \(2, + \infty\).

Step 1: Graphing the function

Plot the function \(f(x)=x^{2}-4x+1\). This is a quadratic function and it will bring up a parabola. The graph will help you visualize the general shape of the function.

Step 2: Finding the vertex

The vertex of a quadratic function \(f(x)=ax^{2}+bx+c\) is given by the formula \(- \frac{b}{2a}\). In this case, \(a=1\) and \(b=-4\), so the x-value of the vertex is \(- \frac{-4}{2*1}=2\). Plug this into the function to get the y-value: \(y=f(2)=(2)^{2}-4*(2)+1= -3\). So the vertex of the function is (2,-3). As \(a=1>0\), we know the parabola opens upwards thus the vertex represents the minimum point of the function.

Step 3: Estimate Intervals

The function \(f(x)=x^{2}-4x+1\) is increasing for \(x>2\) and decreasing for \(x<2\). This is because the slope of the function becomes more positive as x increases and more negative as x decreases from the vertex (2,-3).

Step 4: Summarizing the results

Function \(f(x)=x^{2}-4x+1\) has a minimum at point (2, -3). It is decreasing on the interval \(- \infty, 2\) and increasing on the interval \(2, + \infty\).

Key Concepts

Finding the Vertex of a Parabola

The vertex of a parabola is a critical point that represents the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. It's crucial for understanding the overall shape of the quadratic function.

To find the vertex of a parabola given by a function like \(f(x)=ax^2+bx+c\), you can use the vertex formula \(-\frac{b}{2a}\) to find the x-coordinate. Here, we have the quadratic function \(f(x)=x^2-4x+1\). Let's apply this:

• Identify \(a=1\) and \(b=-4\).
• Calculate the x-coordinate of the vertex: \(-\frac{-4}{2\cdot1}=2\).
• Determine the y-coordinate by plugging it into the function: \(y=f(2)=(2)^2-4\cdot(2)+1=-3\).

The vertex of this parabola, therefore, is at \((2, -3)\). Since \(a\) is positive, our parabola opens upwards, making the vertex the minimum point.

Determining Intervals of Increase and Decrease

After identifying a parabola's vertex, we can determine the intervals where the function increases or decreases. For the function \(f(x)=x^2-4x+1\), we've established that the vertex is at \((2, -3)\). The parabola opens upwards, meaning the function decreases before reaching the vertex and increases after.

To express this as intervals:

• The function is decreasing on the interval \((-fty, 2)\).
• The function is increasing on the interval \((2, +fty)\).

This information is invaluable when analyzing the function's behavior and for understanding its graph. The intervals show us where the function's slope is negative (decreasing) and where it is positive (increasing).

Using Graphing Utilities in Calculus

Graphing utilities are powerful tools in calculus that aid in visualizing functions and their properties. When working with functions like \(f(x)=x^2-4x+1\), a graphing utility can help us plot the function quickly to approximate relative minima or maxima, and identify increasing and decreasing intervals without doing all calculations by hand.

By entering the function into the graphing utility, we can:

• Observe the parabola's shape and determine whether it opens up or down.
• Use the utility's features to locate the vertex rather than calculating it manually.
• Utilize the zoom and trace functions to estimate intervals of increase and decrease accurately.

These features are not just shortcuts; they enhance our understanding of the function's behavior by providing a visual context that complements analytical methods. This combination of graphing technology with traditional calculus techniques enriches the learning experience.