Problem 1 - A Graphical Approach to Precalculus with Limits

Question

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Find the maximum y-value on the graph of \(y=-16 x^{2}+32 x+100\).

Step-by-Step Solution

Answer

The maximum y-value is 116.

Step 1: Identify the Quadratic Function

The given quadratic function is in the form of \[ y = ax^2 + bx + c \]where \( a = -16 \), \( b = 32 \), and \( c = 100 \).

Step 2: Determine the Vertex Formula for y-coordinate

For a parabola represented by the equation \[ y = ax^2 + bx + c \],the y-coordinate of the vertex, which gives the maximum value for a downward opening parabola, can be calculated using the expression \[ y = c - \frac{b^2}{4a} \].

Step 3: Calculate the y-coordinate of the Vertex

Substitute \( b = 32 \) and \( a = -16 \) into the y-coordinate formula: \[y = 100 - \frac{32^2}{4(-16)} \]Simplifying: \[ 32^2 = 1024 \]and \[ 4(-16) = -64 \]. Thus: \[y = 100 - \frac{1024}{-64} = 100 + 16 \]Therefore: \[y = 116 \].

Step 4: Verify the Parabola's Orientation

Since \( a = -16 \) is negative, the parabola opens downwards. This confirms our solution that the vertex provides the maximum y-value.

Key Concepts

Vertex Formula

The vertex formula is a crucial tool in understanding the properties of a quadratic function. In a quadratic equation of the form \( y = ax^2 + bx + c \), the vertex is a point that reveals critical information about the parabola's shape. For a downward-opening parabola, which occurs when \( a \) is negative, the vertex is the highest point on the graph.
To find the y-coordinate of the vertex, indispensable in identifying maximum or minimum values, we employ the formula:

• \( y = c - \frac{b^2}{4a} \)

This formula leverages the coefficients from the quadratic equation, using \( b \) and \( a \), and allows us to calculate the vertex directly. By substituting the appropriate values from the quadratic equation, students can precisely determine the maximum y-value of the graph.
This calculation is critical since it reveals either the peak (maximum) or trough (minimum) depending on whether the parabola opens downwards or upwards.

Parabola

A parabola is a U-shaped graph that represents the line of a quadratic equation. The orientation and position of a parabola are vital for understanding its nature and behavior. When a parabola opens downwards, it means the quadratic function has a negative leading coefficient.

• The general form of a quadratic function is \( y = ax^2 + bx + c \).
• If \( a < 0 \), the parabola opens downwards, indicating the presence of a maximum value.
• The highest point on a downward-opening parabola is its vertex.

The concept of a parabola is essential not only in academic scenarios but also in real-life situations, such as modeling projectile motion. Mastering how to find the vertex and understand parabola orientation aids greatly in solving related problems.

Maximum Value

In a quadratic function, particularly one that forms a downward-opening parabola (when \( a \) is negative), the maximum value is found at the vertex of the parabola. This value represents the peak point of the curve and is the highest attainable y-value on the graph.

• The maximum value is calculated using the vertex formula for the y-coordinate.
• Since the parabola opens downwards, the vertex gives us this peak point.
• For the equation \( y = -16x^2 + 32x + 100 \), the maximum y-value is crucial.

By substituting into the vertex formula, we determined the y-value of 116 as the maximum. Recognizing where the maximum value occurs helps in multiple fields such as physics, economics, and engineering, as it often signifies the optimum point or condition.