Problem 7 - 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. $$ \begin{array}{l}{y=100-x^{2}} \\ {\text { (a) }[-4,4] \text { by }[-4,4]} \\\ {\text { (b) }[-10,10] \text { by }[-10,10]} \\ {\text { (c) }[-15,15] \text { by }[-30,110]} \\ {\text { (d) }[-4,4] \text { by }[-30,110]}\end{array} $$

Step-by-Step Solution

Answer

Option (c) produces the most appropriate graph.

Step 1: Understand the Equation

The given equation is \(y = 100 - x^2\). This is a quadratic function, which represents a parabola opening downward with its vertex at \((0, 100)\).

Step 2: Graph the Equation in Option (a)

For option (a) with the range \([-4, 4]\) for both x and y, plot the graph of \(y = 100 - x^2\). Since \(x^2\) grows quickly, the vertex \((0, 100)\) is outside this viewing rectangle, making it hard to see the curve's peak.

Step 3: Graph the Equation in Option (b)

In option (b) with the range \([-10, 10]\) for both x and y, plot the graph. The vertex \((0, 100)\) is still outside this range, and the graph is not visible.

Step 4: Graph the Equation in Option (c)

For option (c) with \([-15, 15]\) for x and \([-30, 110]\) for y, graph \(y = 100 - x^2\). The vertex \((0, 100)\) is within the viewing rectangle, and the parabola is completely visible, showing the downward opening curve.

Step 5: Graph the Equation in Option (d)

In option (d) with \([-4, 4]\) for x and \([-30, 110]\) for y, the graph will similarly include the vertex \((0, 100)\) but show less of the parabola’s width as compared to option (c).

Step 6: Compare and Decide

Comparing all options, option (c) provides a complete view of the parabola with the vertex visible and enough x-range to show the curve perfectly. This makes it the most appropriate viewing window.

Key Concepts

Parabola Vertex

When discussing a parabola, a key feature to consider is the vertex. The vertex is the highest or lowest point on the graph of a quadratic function, depending on whether the parabola opens upwards or downwards. For the quadratic function given by the equation \(y = 100 - x^2\), the parabola opens downward. The vertex is found using the equation's standard form \(y = ax^2 + bx + c\). In this case, the vertex is \((0, 100)\) because the vertex formula \(x = -\frac{b}{2a}\) shows \(x = 0\). The y-value at this point is 100, which comes directly from the constant term when \(x = 0\). Understanding the position of the vertex is crucial for determining how and where the parabola sits on a coordinate plane.

Viewing Rectangle

When graphing a quadratic function, choosing the correct viewing rectangle can significantly affect the graph's visibility and comprehensiveness. A viewing rectangle is the section of the coordinate plane you choose to focus on when plotting a graph. It is defined by the x and y ranges that you select, influencing what part of the graph is visible. For the function \(y = 100 - x^2\), using a viewing rectangle that contains the vertex \((0, 100)\) and effectively shows the curve’s symmetry is essential. Options like

• Option (c) with x-range ([-15, 15]) and y-range ([-30, 110])
• or Option (d) with x-range (-4, 4) and y-range (-30, 110)

serve well, as they allow you to view the parabola's vertex and a significant part of the curve's width and shape.Selecting an insufficient viewing rectangle may cut off important parts of the graph, such as the vertex or ends, leaving essential features unobserved.

Graphing Calculator

Graphing calculators are valuable tools for students and educators alike when it comes to visualizing mathematical functions. They allow you to input algebraic equations and produce graphical representations on a digital screen. For quadratic functions like \(y = 100 - x^2\), a graphing calculator assists in visualizing how the parabola behaves within different viewing rectangles. This can help determine the most appropriate viewing window to fully capture the graph's essential features, like the vertex and the opening of the parabola. By graphing each suggested rectangle option, one can quickly decide which offers the clearest and most informative view. Using a graphing calculator simplifies the process of guessing or calculating where the vertex lies and how the curve stretches or shrinks across the x-axis and y-axis.

Parabola

A parabola is a U-shaped curve described by a quadratic function, which can open either upwards or downwards depending on the sign of the leading coefficient. In the function \(y = 100 - x^2\), the parabola opens downwards because the coefficient of \(-x^2\) is negative. Essential features of a parabola include its vertex, axis of symmetry, direction of opening, and y-intercept.

• The vertex for this specific function is \((0, 100)\), a crucial point because it represents the maximum height of the parabola.
• The axis of symmetry is the vertical line \(x = 0\),
• and the parabola crosses the y-axis at this vertex point, showing where the maximum value occurs.

Understanding these parabolic features helps in graphically representing the function and predicting its behavior across different graphing scenarios.