Problem 83 - Algebra and Trigonometry

Question

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$ y=-4 x^{2}+20 x+160 $$

Step-by-Step Solution

Answer

The vertex of the given quadratic function is (2.5, 165).

Step 1: Find the Vertex

The vertex \((h, k)\) of a parabola \(y = ax^{2} + bx + c\) is given by the point \((h = -b/2a , k = f(-b/2a)).\) For the given quadratic function \(y = -4x^{2} + 20x + 160\), \(a = -4\), \(b = 20\), \(c = 160\). So, the \(x\)-coordinate of the vertex \(h\) is: \(h = -b/2a = -20 / (2*-4) = 2.5\). Substitute \(h\) into the function to find the \(y\)-coordinate \(k\): \(k = -4(2.5)^2 + 20(2.5) + 160 = 165\). Thus the vertex of the parabola is (2.5, 165).

Step 2: Determine a Viewing Rectangle

A reasonable viewing rectangle on your graphing utility would have a range that includes the vertex and shows the shape of the parabola. As the vertex is at (2.5, 165), you might choose a viewing rectangle with x-values from -5 to 5 and y-values from 0 to 180.

Step 3: Graph the Quadratic Function

Using the above viewing rectangle and the vertex at (2.5, 165), graph the quadratic function. The plot will reveal a downward-opening parabola in line with the negative leading coefficient of the equation.

Key Concepts

Understanding Quadratic Functions

A quadratic function is a polynomial function of the form \( y = ax^2 + bx + c \). It involves squared terms and typically graphs as a parabola, which is a symmetrical curve. The coefficients \( a \), \( b \), and \( c \) define the parabola's orientation and its width. Specifically:

• \( a \): Determines the direction (upward or downward). If \( a > 0 \), the parabola opens upward, and if \( a < 0 \), it opens downward.
• \( b \) and \( c \): Affect the position and form of the curve along the coordinate plane.

In our example, \( y = -4x^2 + 20x + 160 \), the \( a = -4 \) tells us the parabola opens downward.

Utilizing a Graphing Utility

A graphing utility is a powerful tool used by students to visually understand functions and equations. These devices or software allow us to accurately plot functions and analyze their behavior. For quadratic functions:

• The graphing utility helps identify key features such as the vertex, axis of symmetry, and intersection points.
• It enables easy manipulation of the viewing rectangle, which is essential for capturing the important aspects of a curve.

In our problem, adjusting the viewing rectangle ensures the vertex and overall shape of the parabola is clearly visible, aiding in comprehension.

Choosing the Right Viewing Rectangle

The viewing rectangle refers to the window of the graph that is displayed on the graphing utility. It is crucial for properly visualizing the significant features of the function. When selecting a viewing rectangle:

• Ensure it includes key features such as the vertex and intercepts.
• Choose an x-range and y-range that accurately represents changes in the function's curve.

For the example \( y = -4x^2 + 20x + 160 \), a reasonable choice is to set x-values from -5 to 5 and y-values from 0 to 180. This way, the vertex at (2.5, 165) is clearly visible.

Visualizing the Parabola Shape

The shape of the parabola is a distinct feature of quadratic functions. It is characterized by:

• The "U" shape formed by the curve, either opening upwards or downwards.
• The vertex, which is the peak or the lowest point depending on the direction.

With the quadratic function \( y = -4x^2 + 20x + 160 \), its negative leading coefficient \( a = -4 \) indicates it opens downward, forming an arch-like shape. Understanding this shape helps to predict where a function reaches its maximum or minimum, and thus, extend problem-solving skills to other quadratic scenarios.