Problem 72 - Contemporary Precalculus

Question

In Exercises \(69-72,\) graph all four equations on the same screen, using a sufficiently large square viewing window, and answer this question: What is the geometric relationship of graphs (b), (c), and (d) to graph (a)? (a) \(y=x^{2}\) (b) \(y=-x^{2}\) (c) \(y=-\frac{1}{2} x^{2}\) (d) \(y=-2 x^{2}\)

Step-by-Step Solution

Answer

Question: Describe the geometric relationship between graphs (b), (c), and (d) in comparison to graph (a). Answer: Graphs (b), (c), and (d) are all reflections of graph (a) across the x-axis. Additionally, graph (c) is vertically stretched compared to graph (b), while graph (d) is vertically compressed compared to graph (b).

Step 1: Graph equation (a)

Begin with graphing the equation (a) which is y = x^2. This is a basic quadratic function with its parabola opening upwards and its vertex at the origin (0,0).

Step 2: Graph equation (b)

Now, graph the equation (b) which is y = -x^2. This is the reflection of the basic quadratic function y = x^2 across the x-axis, with its parabola opening downwards and its vertex at the origin (0,0).

Step 3: Graph equation (c)

Next, graph the equation (c) which is y = -\frac{1}{2}x^2. This is a vertically stretched reflection of the basic quadratic function y = x^2 across the x-axis, with its parabola opening downwards and its vertex at the origin (0,0). The stretching factor is \frac{1}{2}.

Step 4: Graph equation (d)

Finally, graph the equation (d) which is y = -2x^2. This is a vertically compressed reflection of the basic quadratic function y = x^2 across the x-axis, with its parabola opening downwards and its vertex at the origin (0,0). The compression factor is 2.

Step 5: Determine the geometric relationship

Now that all four equations are graphed, we can observe their geometric relationship. Graphs (b), (c), and (d) are all reflections of the graph (a) across the x-axis. Moreover, graph (c) is vertically stretched compared to graph (b), while graph (d) is vertically compressed compared to graph (b).

Key Concepts

Parabola Graphing

Graphing a parabola is an essential skill when studying quadratic functions. A basic quadratic function is given by the equation \(y = ax^2 + bx + c\), where the graph is a smooth, symmetrical curve called a parabola. The most straightforward example of a quadratic function is \(y = x^2\), where the parabola opens upwards and the vertex—the highest or lowest point on the graph—is located at the origin (0,0).

When graphing parabolas, it's important to recognize the impact that the coefficients \(a\), \(b\), and \(c\) have on the shape and position of the graph. The coefficient \(a\) determines if the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)), as well as the width of the parabola—larger values of \(a\) make it narrower, while smaller values make it wider. To accurately graph a parabola, plotting points and using symmetry can ensure the curve represents the quadratic equation correctly. Sufficiently large square viewing windows can help maintain the parabola's symmetry and proportions for better visualization.

In our exercise example, graphing is done step by step, starting with the simplest form of the quadratic function and progressively introducing modifications.

Reflection Across the X-Axis

Reflection across the x-axis is a powerful concept in the graphing of quadratic functions. When you have a basic parabola such as \(y = x^2\), reflecting it across the x-axis will flip it upside down. Mathematically, this is achieved by multiplying the entire function by -1, resulting in a new function of \(y = -x^2\).

The original and reflected parabolas share the same vertex, which remains fixed at the origin (0,0) in our exercise example. However, the direction of the opening changes—instead of opening upwards, the reflected parabola opens downwards. Knowing how to reflect a graph across the x-axis is critical for understanding symmetry in quadratic functions and how graphs relate to each other geometrically, as seen in our exercise where equations (b), (c), and (d) are all reflections of graph (a).

This transformation does not stretch or compress the graph; it simply inverts it with respect to the x-axis. Recognizing this reflection is key to understanding more complex transformations of parabolas.

Vertical Stretching and Compression

Vertical stretching and compression are transformations that alter the 'width' of a parabola without affecting its direction of opening or axis of symmetry. A vertical stretch makes the parabola narrower, while a vertical compression makes it wider.

These transformations are determined by the absolute value of the coefficient 'a' in the quadratic function \(y = ax^2 + bx + c\). If \( |a| > 1 \), the parabola is vertically compressed, and if \( 0 < |a| < 1 \), it is vertically stretched. In the exercise example, \(y = -\frac{1}{2}x^2\) indicates a vertical stretch since \( |\frac{1}{2}| < 1 \), while \(y = -2x^2\) signifies a vertical compression because \( |2| > 1 \).

Understanding these transformations allows students to predict and graph changes in the parabola's shape just by looking at the equation. In our exercise, the varying coefficients demonstrate the differences in graphs (c) and (d) when compared to graph (b)—highlighting the impact of stretching and compression on a reflected quadratic function.