Problem 8 - Algebra and Trigonometry

Question

\(5-10\) Use a graphing calculator or computer to decide which viewing rectangle \((a)-(\text { d) produces the most appropriate graph }\) of the equation. $$ \begin{array}{l}{y=2 x^{2}-1000} \\ {\text { (a) }[-10,10] \text { by }[-10,10]} \\ {\text { (b) }[-10,10] \text { by }[-100,100]} \\ {\text { (c) }[-10,10] \text { by }[-1000,1000]} \\ {\text { (d) }[-25,25] \text { by }[-1200,200]}\end{array} $$

Step-by-Step Solution

Answer

Option (c) \, \([-10,10] \, \text{by} \, [-1000,1000]\) \, is \, the \, most \, appropriate.

Step 1: Understand the Equation

The given equation is a quadratic function, expressed as \( y = 2x^2 - 1000 \). This describes a parabola that opens upwards and has its vertex at \( (0, -1000) \). To choose an appropriate viewing rectangle, we need to consider the features of this parabola, especially its vertex and the rate at which \( y \) increases as \( x \) moves away from zero.

Step 2: Evaluate First Option

The first option is a viewing rectangle from \([-10, 10]\) on the x-axis and \([-10, 10]\) on the y-axis. Since the vertex is at \( -1000 \), this y-range does not include the vertex, making it a poor choice.

Step 3: Evaluate Second Option

The second option is a viewing rectangle from \([-10, 10]\) on the x-axis and \([-100, 100]\) on the y-axis. Again, this y-range does not cover \(-1000\), hence it is not appropriate either because the vertex still lies outside the view.

Step 4: Evaluate Third Option

The third option is a viewing rectangle from \([-10, 10]\) on the x-axis and \([-1000, 1000]\) on the y-axis. This y-range includes the vertex \(-1000\), allowing the vertex of the parabola to be visible in the graph.

Step 5: Evaluate Fourth Option

The fourth option is a viewing rectangle from \([-25, 25]\) on the x-axis and \([-1200, 200]\) on the y-axis. This range also includes the vertex \(-1000\) and extends beyond \[-25,25\] for the x-range, providing a broader view of the parabola's shape.

Step 6: Choose the Best Option

Among the options, (c) with the y-range \([-1000, 1000]\) is both inclusive of the vertex and a balanced choice in terms of viewing the parabola clearly. This makes it the most appropriate graph.

Key Concepts

Understanding a Parabola

A parabola is a symmetric curve that is shaped like an open bowl. It is the graphical representation of a quadratic function, typically expressed as \( y = ax^2 + bx + c \). In this general form, the variable \( a \) determines whether the parabola opens upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)).
For the given equation \( y = 2x^2 - 1000 \), the parabola opens upwards because the coefficient of \( x^2 \) is positive. Each parabola is defined by its vertex, which is the point where the curve changes direction, giving it a U-shape. The symmetry of a parabola means that it looks the same on both sides of its vertical axis (the line passing through the vertex).
Parabolas have key features such as the vertex, axis of symmetry, and intercepts, which make them predictable and easy to plot using a basic understanding of their equation.

Choosing a Viewing Rectangle

A viewing rectangle is the portion of the coordinate plane that you choose to display when graphing a function on paper or a screen. It is essential to select an appropriate viewing rectangle to ensure that all critical parts of the graph are visible. This helps to better understand the behavior and features of the function.
In the task, different viewing rectangles were provided to evaluate which would offer the most informative graph of the given parabola. The chosen rectangle affects what is seen on the graph, such as its vertex, intercepts, and the general shape of the parabola.
To properly view our parabola, which has a vertex at \( y = -1000 \), the viewing rectangle should cover the y-values as low as \( -1000 \) and beyond. This inclusion allows the vertex to be visible, providing complete visibility of the parabola’s features. Hence, the right choice ensures a clear depiction of the quadratic function on a graphing platform.

Using a Graphing Calculator

A graphing calculator is a powerful tool used to visualize mathematical equations, especially when dealing with functions like parabolas. Unlike basic calculators, graphing calculators can plot complex graphs and provide visual aid for understanding algebraic concepts. They allow for manipulation and observation of changes when variables in functions are adjusted.
When using a graphing calculator for graphing parabolas, it’s essential to input the correct settings, especially in choosing a suitable viewing rectangle. This approach ensures that critical points such as the vertex and intercepts are visible. Additionally, it enables students to experiment by adjusting the window settings to observe how the parabola behaves under different conditions.
By visualizing the parabola \( y = 2x^2 - 1000 \), students gain insight into its shape, scale, and where the vertex lies on the graph. This process helps bring abstract equations to life, making them easier to comprehend and analyze.

Focusing on the Vertex of a Parabola

The vertex of a parabola is its most crucial point, marking the minimum or maximum value on the graph, depending on the parabola's orientation. For the equation \( y = 2x^2 - 1000 \), the vertex is at \( (0, -1000) \).
The vertex provides valuable information: it's where the parabola changes direction, and it's often the lowest point when the parabola opens upwards, or the highest when it opens downwards. Knowing the vertex is vital for graphing because it serves as a reference point from which the rest of the curve is plotted.
To find the vertex algebraically in the standard form \( y = ax^2 + bx + c \), you can use the formula \( x = -\frac{b}{2a} \) to determine the x-coordinate of the vertex. Substituting this x-value back into the equation will provide the y-coordinate. Remembering this approach is a helpful way to confirm your graphical results obtained from a calculator or plotting manually.