Problem 40 - College Algebra with Modeling and Visualization

Question

Modeling Decide if the situation could be modeled by a one-to-one function. The height \(y\) of a stone thrown upward after \(x\) seconds

Step-by-Step Solution

Answer

No, the situation cannot be modeled by a one-to-one function.

Step 1: Understanding the Scenario

The scenario involves a stone being thrown upward, and we want to determine if the height of the stone at any given time can be described as a one-to-one function. A one-to-one function is where each output value corresponds to exactly one input value.

Step 2: Analyze the Height Function

When a stone is thrown upward, its height over time follows a quadratic function due to gravity, typically of the form \( y = -ax^2 + bx + c \), where \(a, b, \text{and} c\) are constants, and \(a > 0\). This means the graph is a parabola that opens downward.

Step 3: Determine Function Behavior

Since the height function is quadratic, it has a vertex, and the stone will reach a maximum height before falling back down. Thus, there will be two different time values (before and after the peak) for the same height during its trajectory.

Step 4: Analyse One-to-One Condition

For a function to be one-to-one, each height value should correspond to exactly one time value. However, because the stone reaches the same height twice (once on the way up and once on the way down), the function is not one-to-one.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree is two. It is typically represented in the form \( y = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). This form is known as the standard form of a quadratic function. These functions graph as a curve known as a parabola. Quadratic functions are used in various applications, such as calculating areas, projectile motion (like the one in our problem), and even in optimizing business profits.

Quadratic functions have several key features that make them distinct:

• They are symmetric with respect to a vertical line, known as the axis of symmetry.
• The direction of the parabola (upward or downward) is determined by the leading coefficient \(a\).
• The highest or lowest point on the graph, depending on the parabola's direction, is called the vertex.

Exploring these characteristics can help us understand events and behaviors that involve quadratic patterns, such as the height of the stone over time in projectile motion.

Parabola

A parabola is the graph of a quadratic function, exhibiting a smooth, symmetrical curve. Depending on the sign of the leading coefficient \(a\), it can open either upward or downward. In the problem being discussed, the function is of the form \( y = -ax^2 + bx + c \), indicating a downward-opening parabola due to the negative \(a\) value, which represents gravity's influence.

Some noteworthy properties of parabolas include:

• The vertex, which is the peak or trough of the parabola, is the point where the axis of symmetry intersects the parabola.
• The axis of symmetry is a vertical line that runs through the vertex, dividing the parabola into two mirror-image halves.
• The focus and directrix, specialized lines and points, help define the curve's shape in a geometric context.

When analyzing a stone thrown upwards, the downward bowl shape of the parabola illustrates how its height increases to a maximum point before decreasing again due to gravity.

Vertex

The vertex of a parabola, particularly with a quadratic function like \( y = ax^2 + bx + c \), is the highest or lowest point, depending on the orientation of the parabola. It represents the maximum height in the context of a projectile motion like our stone's path.

To find the exact position of the vertex, you can use the vertex formula \( x = \frac{-b}{2a} \) for the \(x\)-coordinate of the vertex. Substituting this \(x\)-value back into the quadratic function will provide you with the corresponding \(y\)-coordinate.

Characteristics of the vertex include:

• It is at the intersection of the parabola and its axis of symmetry.
• On a downward-opening parabola, it marks the maximum point, while on an upward-opening parabola, it marks the minimum.
• In a physical context, like the path of a stone in air, the vertex represents the instance of maximum height.

Understanding the position and significance of the vertex can help in predicting and analyzing the trajectory of objects under the influence of gravity.