Problem 7 - 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. \(g(x)=-x^{2}+1\)

Step-by-Step Solution

Answer

The sequence of transformations to go from \(f(x)=x^{2}\) to \(g(x)=-x^{2}+1\) is a reflection over the x-axis followed by a shift upward by 1. After drawing, verification with a graphical utility ensures correct interpretation.

Step 1: Identify the Reflection

The function \(f(x)=x^{2}\) has been transformed into \(g(x)=-x^{2}\). The negative sign before the square function \(x^{2}\) indicates the reflection of function \(f(x)=x^{2}\) about the x-axis to get the graph of \(-x^{2}\).

Step 2: Identify the Vertical Shift

After the reflection transformation, the function will be further transformed from \(-x^{2}\) to \(-x^{2}+1\). This means the function has been shifted upward by 1. This is a vertical shift.

Step 3: Sketch the Graph of g

Start with the standard graph of \(f(x)=x^{2}\), a U-shaped parabola opening upwards with the vertex at the origin. Apply transformations identified in step 1 and 2. First reflect the graph of \(f(x)=x^{2}\) over the x-axis to get \(-x^{2}\), then shift the graph of \(-x^{2}\) upward by 1 unit to get \(-x^{2}+1\). The final graph will be a U-shaped parabola opening downwards, with the vertex at (0,1).

Step 4: Verify with a Graphing Utility

Plot both \(f(x)=x^{2}\) and \(g(x)=-x^{2}+1\) on a graphing calculator or software (e.g., Desmos, GeoGebra, etc.) to verify the accuracy. The graph of \(g(x)\) should appear as a vertically shifted, reflection of the original function \(f(x)\).

Key Concepts

Reflection about the x-axis

The concept of reflection about the x-axis is an essential transformation when dealing with functions. To understand this, imagine looking at yourself in a pond – your reflection is upside down. A similar transformation occurs with graphs. When a function, such as the simple quadratic function \(f(x)=x^{2}\), undergoes reflection about the x-axis, it flips over the x-axis.
This transformation changes any positive output value of the function to its negative counterpart and vice versa.

• The function \(f(x)=x^{2}\) has all positive outputs because it's a parabola opening upwards.
• By introducing a negative sign, as in \(g(x)=-x^{2}\), the graph is reflected upside down over the x-axis.
• This creates a parabola opening downwards.

After performing this reflection, each point on the original parabola \((x, y)\) maps to a new point \((x, -y)\) on the reflected graph. This is a fundamental transformation that alters the orientation of the graph.

Vertical shift

After reflecting the graph, we apply a vertical shift to transform \(g(x)\). A vertical shift is a type of transformation that moves a graph up or down on the coordinate plane.
In this problem, after reflecting \(f(x)=x^{2}\) to \(-x^{2}\), we add 1 to the entire function, resulting in \(g(x)=-x^{2} + 1\).

• Addition of a constant to the function causes a vertical shift.
• In our case, \(+1\) moves every point on \(-x^{2}\) upward by 1 unit.

Now, the vertex of the parabola, which was at the origin \((0, 0)\) after reflection, moves to \((0, 1)\).
This means all points of the graph have been lifted up by one unit, keeping the shape identical.

Parabolic graph transformations

Parabolic graph transformations include a variety of operations such as reflections, vertical shifts, stretches, and compressions. When working with transformations, it is important to start with the base function.
In our exercise, we begin with \(f(x)=x^{2}\), which is a basic parabola with a vertex at the origin and symmetric about the y-axis.

• The transformation sequence for \(g(x)=-x^{2} + 1\) begins by reflecting over the x-axis, changing the parabola's orientation from upward to downward.
• Next, we shift the entire graph upward by 1 unit.

Graph transformations can be visualized as a set of operations that modify the appearance or position of a graph, while retaining its characteristic parabola shape.
The transformed graph \(g(x)\), with vertex at \((0, 1)\), maintains the symmetry and broad U-shape typical of quadratics, indicating a successful transformation while showing the effects of the reflection and vertical shift.