Problem 1 - College Algebra and Calculus: An Applied Approach

Question

Describe the sequence of transformations from \(f(x)=x^{2}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=x^{2}-4\)

Step-by-Step Solution

Answer

The sequence of transformation is a vertical downward shift of the parent function \(f(x)=x^{2}\) by 4 units to get the function \(g(x)=x^{2}-4\). The graph of the function \(g(x)\) would be a parabola with vertex at (0, -4). The transformation and the graph sketch can be verified using a graphing utility.

Step 1: Identifying the Transformation

The original function is \(f(x) = x^{2}\) and the transformed function is \(g(x) = x^{2} - 4\). Comparing these two functions, the main transformation that has occurred is a vertical shift. The '-4' in \(g(x)\) is affecting the constant term in the function, which corresponds to a shift along the y-axis. Since '-4' is subtracted, it means the function \(f(x)\) has been shifted downwards by 4 units to result in \(g(x)\).

Step 2: Sketching the Graph

First, sketch the graph of the original function \(f(x) = x^{2}\). This will be a parabola opening upwards, touching the origin (0,0). Now for \(g(x)\), keep the same shape of the parabola, but shift it down by 4 units. This means the vertex of the parabola will now be at the point (0, -4).

Step 3: Verification using Graphing Utility

To ensure the correctness of our transformation and the resulting sketch, use a graphing utility like a graphing calculator or an online tool to plot the function \(g(x)=x^{2}-4\). The graph shown should match your sketch - a downward opening parabola, with its vertex at the point (0, -4).

Key Concepts

Vertical Shift

The transformation from the function \(f(x) = x^{2}\) to \(g(x) = x^{2} - 4\) includes a vertical shift. A vertical shift involves moving the graph of a function up or down along the y-axis without altering its shape. This happens by adding or subtracting a constant from the function. In our case:

• The constant "-4" is subtracted from \(x^{2}\).
• This means the graph moves down by 4 units.
• Each point on the original graph of \(f(x) = x^{2}\) is lowered by 4 units.

You can visualize this shift by imagining the entire parabola being pushed down the y-axis until the new vertex is at (0, -4). This transformation keeps the parabola's shape and width but changes its vertical location on the coordinate plane.

Graph Sketching

To sketch the graph of \(g(x)=x^{2}-4\), start with the basic parabola \(f(x)=x^{2}\). This is a standard upward-opening parabola with the vertex at the origin. The steps to sketch \(g(x)=x^{2}-4\) are simple:

• Draw the graph of \(f(x)=x^{2}\), with its smooth U-shape and vertex at (0,0).
• Now, apply the vertical shift: move the entire graph down by 4 units.
• The new vertex should land directly at (0, -4).

This adjustment results in a parabola that looks identical to \(f(x)=x^{2}\), but its "starting" point (vertex) is lower. The arms of the parabola extend upwards and outwards the same way, maintaining symmetry along the y-axis.

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic function. It has key characteristics:

• Vertex: The highest or lowest point, depending on the direction it opens.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves.
• Direction: Can open upwards or downwards depending on the coefficient of the square term.

For the functions \(f(x)=x^{2}\) and \(g(x)=x^{2}-4\):

• Both open upwards because the coefficient of \(x^{2}\) is positive.
• The vertex for \(f(x)=x^{2}\) is at the origin (0,0).
• For \(g(x)=x^{2}-4\), after the vertical shift, the vertex is at (0, -4).
• The axis of symmetry for both is the y-axis, which means every point on the parabola has a corresponding point directly on the opposite side, equidistant from the axis.

Understanding these details makes it easier to predict and sketch the graph of quadratic functions after transformations.