Problem 60 - College Algebra with Modeling and Visualization

Question

Use f(x) to determine verbal, graphical and numerical representations. For the numerical representation use a table wish \(x=-2,-1,0,1,2\) Evaluate \(f(2).\) $$ f(x)=x^{2}-1 $$

Step-by-Step Solution

Answer

Numerically and graphically it shows a parabola; \(f(2) = 3\).

Step 1: Understanding the Function

The function given is a quadratic function, written as \(f(x) = x^2 - 1\). This function represents a parabola that opens upwards.

Step 2: Numerical Representation

We will evaluate the function at different values of \(x\), specifically for \(x = -2, -1, 0, 1, 2\), and create a table to represent these values.\[\begin{array}{|c|c|}\hlinex & f(x) \\hline-2 & (-2)^2 - 1 = 3 \-1 & (-1)^2 - 1 = 0 \0 & (0)^2 - 1 = -1 \1 & (1)^2 - 1 = 0 \2 & (2)^2 - 1 = 3 \\hline\end{array}\]

Step 3: Verbal Representation

The function \(f(x) = x^2 - 1\) represents a parabola that opens upwards with its vertex at the point \( (0, -1) \). The function is symmetric about the y-axis and has roots at \(x = -1\) and \(x = 1\).

Step 4: Graphical Representation

To graph the function, plot the points from the numerical table: \((-2, 3), (-1, 0), (0, -1), (1, 0), (2, 3)\). Connect these points in a smooth, upward-opening parabola. This will help in visualizing the curve and understanding its shape.

Step 5: Evaluating f(2)

To find \(f(2)\), substitute \(x = 2\) into the function:\[ f(2) = 2^2 - 1 = 4 - 1 = 3. \]Thus, \(f(2) = 3\).

Key Concepts

Numerical Representation

Numerical representation involves expressing the function using specific numerical values. For the quadratic function \(f(x) = x^2 - 1\), we need to calculate its values for different \(x\) values to fully understand its numerical representation. By substituting the given values \(x = -2, -1, 0, 1, 2\) into the function, we generate corresponding \(f(x)\) values. This produces a table of results:

• \((-2, 3)\) - Substitute \(-2\) to get \(f(-2) = (-2)^2 - 1 = 4 - 1 = 3\).
• \((-1, 0)\) - Substitute \(-1\) to get \(f(-1) = (-1)^2 - 1 = 1 - 1 = 0\).
• \((0, -1)\) - Substitute \(0\) to get \(f(0) = (0)^2 - 1 = 0 - 1 = -1\).
• \((1, 0)\) - Substitute \(1\) to get \(f(1) = (1)^2 - 1 = 1 - 1 = 0\).
• \((2, 3)\) - Substitute \(2\) to get \(f(2) = (2)^2 - 1 = 4 - 1 = 3\).

This table helps us see how the function behaves numerically over the specified range of \(x\) values.

Graphical Representation

A graphical representation of a function offers a visual insight into the function's behavior. For the function \(f(x) = x^2 - 1\), we plot the points derived from the numerical evaluation, which were \((-2, 3)\), \((-1, 0)\), \((0, -1)\), \((1, 0)\), and \((2, 3)\). When these points are marked on a graph, they form a parabolic curve. This parabola opens upwards, as its quadratic term \(x^2\) is positive.

• The vertex of the parabola, the point \((0, -1)\), is the lowest point on the graph.
• The graph is symmetric about the y-axis, meaning it mirrors itself on either side of this axis.
• The function touches the x-axis at the roots, \(x = -1\) and \(x = 1\), confirming where the graph intersects the x-axis.

Drawing a smooth curve through these points completes the graph, showing a parabola with its vertex at \((0, -1)\) and extending upwards on either side.

Verbal Representation

Verbal representation translates the mathematical and graphical characteristics of a function into words. For the quadratic function \(f(x) = x^2 - 1\), we can describe it verbally in the following ways:

• The graph is a parabola that opens upwards because the coefficient of \(x^2\) is positive.
• The parabola has a vertex at point \((0, -1)\), which is its minimum value.
• It is symmetric about the y-axis, showing a balanced shape on both sides.
• The function has roots at \(x = -1\) and \(x = 1\), indicating where it crosses the x-axis.
• This means that for any \(x\) less than \(-1\) or greater than \(1\), the function's values are positive, while between these points, the values dip below the x-axis.

Verbal expression helps encapsulate the function's character, making it relatable and easier to understand in a broader context.