Problem 53 - 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 \(f(2).\) $$ f(x)=x^{2} $$

Step-by-Step Solution

Answer

\( f(2) = 4 \)

Step 1: Understand the function verbally

The function given is \( f(x) = x^2 \), which means for every input \( x \), the output is the square of \( x \). This is a quadratic function, representing a parabola that opens upwards. Its vertex is at the origin \((0,0)\).

Step 2: Create a table for graphical and numerical representation

We'll make a table to plug in the values \( x = -2, -1, 0, 1, 2 \) and evaluate \( f(x) \) for each.\(\begin{array}{|c|c|}\hline x & f(x) = x^2 \ \hline -2 & 4 \ \hline -1 & 1 \ \hline 0 & 0 \ \hline 1 & 1 \ \hline 2 & 4 \ \hline\end{array}\)This table helps visualize the symmetry and shape of the parabola.

Step 3: Graphical representation

Using the table, plot the points \((-2,4), (-1,1), (0,0), (1,1), (2,4)\) on a coordinate plane. Connect these points to see a U-shaped curve, illustrating how \( f(x) = x^2 \) forms a parabola.

Step 4: Evaluate \(f(2)\)

Substitute \( x = 2 \) into the function, \( f(x) = x^2 \), to find \( f(2) \).\[ f(2) = 2^2 = 4 \] Thus, \( f(2) = 4 \), as calculated from the function.

Key Concepts

Graphical Representation

The graphical representation of a quadratic function like \( f(x) = x^2 \) involves plotting points on a coordinate plane to visualize its shape. To form the graph, start by selecting a set of \( x \) values, such as \( -2, -1, 0, 1, \) and \( 2 \), and calculate their corresponding \( f(x) \) values using the function. For example, \( f(-2) = (-2)^2 = 4 \) and \( f(2) = 2^2 = 4 \). This gives us points such as \(( -2, 4)\) and \((2, 4)\).
Once you have these pairs of \( (x, f(x)) \), plot them on the graph. Connect the dots smoothly to reveal the shape of the curve. The graph of \( f(x) = x^2 \) is a symmetrical U-shape, which we call a parabola. The vertex, or the lowest point, of this curve is at the origin, \((0,0)\), where the function value is at its minimum in this case. "Graphical representation" provides a powerful visual to understand not just where the function increases or decreases, but how it behaves with respect to its symmetry and key features.

Numerical Representation

The numerical representation is complemented by building a table of values, which basically lists out pairs of input values \( x \) and their corresponding output \( f(x) \). Let's consider the table:

• \( x = -2 \), \( f(x) = 4 \)
• \( x = -1 \), \( f(x) = 1 \)
• \( x = 0 \), \( f(x) = 0 \)
• \( x = 1 \), \( f(x) = 1 \)
• \( x = 2 \), \( f(x) = 4 \)

Constructing this table involves substituting each \( x \) into the function \( f(x) = x^2 \). This numerical data gives a clear snapshot of how the function outputs, doubles or stays the same depending on input. With quadratic functions specifically, you can observe symmetry around the vertex, where the function repeats values for negative and positive \( x \) values equidistant from zero, showcasing a key feature of parabolic shapes.

Parabolic Shape

Quadratic functions exhibit a parabolic shape, characteristic of their graphical output. The equation \( f(x) = x^2 \) is a basic form of a quadratic function, opening upwards to create a U-shaped curve. This parabola has certain unique features:

• **Vertex**: The turning point or the minimum point of the graph is called the vertex. For \( f(x) = x^2 \), the vertex is at the origin \((0,0)\).
• **Axis of Symmetry**: A vertical line that runs through the vertex splits the parabola into mirror images. For this function, the axis is the y-axis, \( x = 0 \).
• **Direction**: Since the coefficient of \( x^2 \) is positive in \( f(x) = x^2 \), the parabola opens upwards. If it were negative, the parabola would open downwards.

Understanding these features not only helps in recognizing the graph of a quadratic function but also in solving complex equations by predicting behavior and key intersections.

Evaluating Functions

Evaluating functions involves substituting specific values into the function to find outputs. For \( f(x) = x^2 \), evaluating is straightforward. Let's say you want to find \( f(2) \). Substitute \( 2 \) for \( x \) in the equation:
\[ f(2) = 2^2 = 4\]This tells us that when \( x = 2 \), \( f(x) = 4 \). This process helps understand how the function transforms input values to output values across its domain.
It allows us to predict real-world scenarios if this function were to model a physical situation, such as calculating area or projectile movements, by simply evaluating these functions for relevant inputs.