Problem 27 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{array}{l}f(x)=x^{2} \\\g(x)=-x^{2}\end{array}$$

Step-by-Step Solution

Answer

To sketch the graph of f(x) = x^2 and g(x) = -x^2 on the same axes, first draw the parabola of f(x) opening upwards with a vertex at (0,0) and key points such as (1,1), (-1,1), (2,4), and (-2,4). Next, observe that g(x) is a reflection of f(x) over the x-axis, so sketch the graph of g(x) opening downwards with the same vertex at (0,0) and reflected key points such as (1,-1), (-1,-1), (2,-4), and (-2,-4). Ensure that both graphs share the origin as a common vertex and are symmetric with respect to the y-axis.

Step 1: Sketch the graph of f(x)

To sketch the graph of f(x)=x^2, recall that this is a standard quadratic function, with the parabola opening upwards and vertex at the origin (0,0). The graph will be axis-symmetric with respect to the y-axis. You can plot additional key points such as (1,1), (-1,1), (2,4), and (-2,4) to better visualize the curve.

Step 2: Understand the transformation

Observe that g(x)=-x^2 is a horizontal reflection of f(x)=x^2. This means that the graph of g(x) will be a reflection of the graph of f(x) over the x-axis. Points on the graph of f(x) having coordinates (x, y) will transform to new points with coordinates (x, -y) on the graph of g(x).

Step 3: Sketch the graph of g(x)

Now, sketch the graph of g(x)=-x^2 on the same axis as f(x)=x^2. Reflect each key point from the graph of f(x) over the x-axis. For example, if we have a point (1,1) on the graph of f(x), we will have a point (1,-1) on the graph of g(x). Similarly, reflect other key points such as (-1, 1) to (-1, -1) and (2, 4) to (2, -4), and so on.

Step 4: Double-check your work

Ensure that both the graphs of f(x)=x^2 and g(x)=-x^2 are on the same axes, with the parabola of f(x) opening upwards and the parabola of g(x) opening downwards. The graph of g(x) should be the reflection of f(x) over the x-axis. Both parabolas should share a common vertex at the origin (0,0) and be symmetric with respect to the y-axis. By following these steps, you have successfully sketched the graphs of f(x) and g(x) on the same axes using transformation techniques.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest exponent of the variable is 2. These functions play a major role in many areas of mathematics and are often represented by the general form:

• \( f(x) = ax^2 + bx + c \)

In this form, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\), because if \(a\) were zero, the function would not be quadratic. The graph of a quadratic function is always a parabola, which is a U-shaped curve. Quadratic functions can vary based on:

• The direction the parabola opens (upward if \(a > 0\), downward if \(a < 0\))
• The width of the parabola (wider as \(|a|\) decreases, narrower as \(|a|\) increases)
• The position of the vertex, determined by the values of \(b\) and \(c\)

Understanding quadratic functions is crucial as they lay the foundation for analyzing kindred mathematical phenomena and real-world applications.

Parabola

A parabola is the geometric shape that represents a quadratic function graph. This symmetrical curve can open either upwards or downwards, characterized by its key features:

• Vertex: The point where the parabola changes direction. In the function \(f(x) = ax^2 + bx + c\), the vertex equation is \(-\frac{b}{2a}\) for x-coordinate.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
• Direction: Determined by the sign of \(a\). Positive \(a\) means the parabola opens upwards, while negative \(a\) indicates it opens downwards.
• Focus and Directrix: These are often used in more advanced analyses, defining the set points from which the parabola is equidistant.

In graphing, understanding these components helps plot the parabola accurately and predict the function's behavior over different intervals. When practicing with problems like sketching graphs, the symmetrical nature of parabolas provides a visual method to cross-check the correctness of your sketch.

Reflection Transformation

A reflection transformation changes the position of an object by flipping it over a specified axis, creating a mirror image. In the context of the quadratic function exercise given, a reflection occurs when transforming the graph of \(f(x) = x^2\) to \(g(x) = -x^2\).

• The reflection takes place over the x-axis.
• Graphically, it means every point \((x, y)\) on \(f(x)\) will correspond to \((x, -y)\) on \(g(x)\).

This transformation alters not only the position of the original graph but also changes the direction in which the parabola opens:

• The original parabola opens upwards while the reflected one opens downwards.

To implement this, sketch the original parabola, identify key points, and systematically reflect these points across the x-axis. This offers a clear, structured method for transforming quadratic graphs and enhances understanding of broader geometric transformations.