Problem 29 - Algebra and Trigonometry

Question

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(f(x)=8 x-x^{2}\) (a) \([-5,5]\) by \([-5,5]\) (b) \([-10,10]\) by \([-10,10]\) (c) \([-2,10]\) by \([-5,20]\) (d) \([-10,10]\) by \([-100,100]\)

Step-by-Step Solution

Answer

Option (c) is the most appropriate: \\([-2, 10] \\\text{ by } [-5, 20]\\).

Step 1: Identify the Function's Type and Key Features

The given function is a quadratic function, specifically a downward-opening parabola, because its highest degree term \(x^2\) has a negative coefficient. The general form of the function is \(f(x) = 8x - x^2\). This means it has a maximum point since the coefficient of \(x^2\) is negative. The vertex form can help find this maximum point.

Step 2: Find the Vertex of the Parabola

To find the vertex of the quadratic function \(f(x) = 8x - x^2\), use the formula for the vertex of \(ax^2 + bx + c\), which is given by \(x = -\frac{b}{2a}\). Here, \(a = -1\) and \(b = 8\), so \(x = -\frac{8}{2 \cdot (-1)} = 4\). Substitute \(x = 4\) back into the function to find \(f(4) = 8 \times 4 - 4^2 = 32 - 16 = 16\). Thus, the vertex is \((4, 16)\). This is the maximum point of the function.

Step 3: Determine the Behavior at the Extremes

Since the leading coefficient \(a = -1\) is negative, the arms of the parabola extend downward to infinity. The y-intercept occurs when \(x=0\), which is \(f(0) = 8 \times 0 - 0^2 = 0\). Thus, the parabola passes through the origin and extends infinitely in the negative direction as \(x ightarrow \pm\infty\).

Step 4: Examine Each Viewing Rectangle

Evaluate how well each viewing rectangle captures the shape and vertex of the parabola. - (a) \([-5, 5] \text{ by } [-5, 5]\) - This range will capture the y-intercept \((0, 0)\) but not fully capture the vertex \((4, 16)\) as it's outside the y-range.- (b) \([-10, 10] \text{ by } [-10, 10]\) - This larger range contains the vertex and goes beyond \(x = 10\) but not high enough to include all of \(y = 16\).- (c) \([-2, 10] \text{ by } [-5, 20]\) - This range includes the vertex \((4, 16)\) and covers a larger portion of the parabola, capturing the maximum point and going slightly beyond the x-range needed.- (d) \([-10, 10] \text{ by } [-100, 100]\) - This range captures the vertex but may excessively expand the viewing window, making details less visible due to empty space.

Step 5: Select the Most Appropriate Viewing Rectangle

Based on the analysis, option (c) \([-2, 10] \text{ by } [-5, 20]\) is the most appropriate rectangle as it includes the vertex, a sufficient range of x-values, and the entire y-range necessary to visualize the parabola's vertex and general shape.

Key Concepts

Parabola Vertex

In quadratic functions, the vertex is a crucial point that indicates either the peak or the trough of the parabola. For the function \(f(x) = 8x - x^2\), the vertex needs to be found to properly understand the parabola's highest point, as the coefficient of \(x^2\) is negative, signaling that the parabola opens downward.
The vertex can be calculated using the formula \(x = -\frac{b}{2a}\), where \(a = -1\) and \(b = 8\). Substituting these values, we find the vertex at \(x = 4\). By evaluating \(f(4)\), substituting back into the function, we get \(f(4) = 16\). Thus, the vertex of this parabola is at the point \((4, 16)\).
Recognizing the vertex position is key to drawing the graph and understanding where the function achieves its maximum value.

Graphing Functions

When graphing quadratic functions like \(f(x) = 8x - x^2\), ensuring the graph's viewing window is appropriately set is essential for capturing the significant features of the function. Each specified viewing rectangle influences the clarity and accuracy with which you can observe key parts of the graph.
Option (a)

• Rectangle: \([-5, 5]\) by \([-5, 5]\).
• This range captures the y-intercept \((0, 0)\), but since the vertex \((4, 16)\) lies outside this y-range, it won't be adequately represented.

Option (b)

• Rectangle: \([-10, 10]\) by \([-10, 10]\).
• While this covers a broader x-range, the y-range does not reach up to the vertex's maximum y-value of 16.

Option (c) stands out:

• Rectangle: \([-2, 10]\) by \([-5, 20]\).
• This viewing window includes the vertex and displays both the x-range and enough of the y-axis to see the full shape and the vertex of the parabola.

Option (d)

• Rectangle: \([-10, 10]\) by \([-100, 100]\).
• This overly extensive viewing range dilutes the detail, showing too much unused space.

Ultimately, option (c) provides the best view to analyze the parabola's characteristics clearly.

Function Behavior Analysis

Understanding the behavior of a function like \(f(x) = 8x - x^2\) is essential to predict its graph's shape and direction. For functions classified as quadratic, their behavior is largely determined by the leading coefficient of \(x^2\). Here, the negative coefficient indicates a downward-opening parabola, meaning the function has a maximum point, which corresponds to the vertex.
Identifying other key points helps describe the function further:

• The y-intercept occurs at \(x = 0\), evaluated as \(f(0) = 0\).
• The vertex, located at \((4, 16)\), represents the function's peak or maximum value.
• As \(x\) approaches plus or minus infinity, the function value heads toward negative infinity due to the downward opening arms of the parabola.

An understanding of these behaviors provides deeper insight into the parabola's shape and how it will be portrayed when graphed, especially in choosing an appropriate viewing rectangle.