Question

The number of horsepower \(H\) required to overcome wind drag on an automobile is approximated by \(H(x)=0.002 x^{2}+0.005 x-0.029, \quad 10 \leq x \leq 100\) where \(x\) is the speed of the car (in miles per hour). (a) Use a graphing utility to graph the function. (b) Rewrite the horsepower function so that \(x\) represents the speed in kilometers per hour. [Find \(H(x / 1.6) .]\) Identify the type of transformation applied to the graph of the horsepower function.

Short answer

Part (a): The graph is a parabola opening upward; graph the function through a graphing utility with the domain \(10 \leq x \leq 100\). Part (b): The modified function for speed in km/hr is \(H(x/1.6) = 0.002(\frac{x}{1.6})^{2} + 0.005(\frac{x}{1.6}) - 0.029\), and the modification is a horizontal stretch of the graph of the original horsepower function by a factor of 1.6.

Step-by-step solution

Understand the function

The given function \(H(x) = 0.002x^{2} + 0.005x - 0.029\) is a quadratic function. This function represents the horsepower required to overcome wind drag on a car as a function of its speed \(x\) (in miles per hour).

Graphing the function (Part a)

This step involves using graphing utility to graph the function. It is not possible to demonstrate the graphing process in this text format. However, to do this on a graphing calculator or software, input the function \(H(x) = 0.002x^{2} + 0.005x - 0.029\) and specify the domain as \(10 \leq x \leq 100\).

Function transformation (Part b)

Since 1 mile = 1.6 kilometers, to convert the speed \(x\) in miles per hour to kilometers per hour, divide \(x\) by 1.6. Replace \(x\) in the original function with \(x/1.6\) to get \(H(x/1.6) = 0.002(\frac{x}{1.6})^{2} + 0.005(\frac{x}{1.6}) - 0.029\). This transformation is a horizontal stretch of the graph of the original horsepower function by a factor of 1.6, to account for the change in units from miles to kilometers.

Key concepts

Graphing Quadratic Functions

Understanding the shape and properties of quadratic functions is crucial in subjects like Physics and Economics, where they often appear. The given equation, \(H(x) = 0.002x^{2} + 0.005x - 0.029\), showcases a classic example of a quadratic function, describing the horsepower needed to overcome wind drag based on a car's speed.

When graphing a quadratic function, such as \(H(x)\), we often see a parabola. This U-shaped curve can open upwards or downwards, determined by the sign of the coefficient of the \(x^2\) term – a positive coefficient, like 0.002 here, means the parabola opens upwards. For an effective graph, one needs to plot sufficient points to capture the curve, paying special attention to the vertex, which is the peak or valley of the parabola, and the roots, if there are any, where the function intersects the x-axis.

Key Features of a Parabola

• Vertex: The highest or lowest point on the graph.
• Axis of Symmetry: A vertical line through the vertex, dividing the parabola into two mirror images.
• Intercepts: Points where the parabola crosses the x-axis and y-axis.

Using a graphing utility, plotting this function between the given domain of \(10 \) and \(100\) miles per hour, helps visualize how the required horsepower changes with speed.

Function Transformation

Function transformation involves altering a function's graph in various ways – including shifts, stretches, and reflections – to produce a new graph. The transformation of the given horsepower function due to a change in units captures a fundamental concept: horizontal stretching.

The transformation \(H(x/1.6)\) applied to our function scales the x-values by a factor of 1.6 to convert miles per hour to kilometers per hour. This is known as a horizontal stretch because all points on the graph move further apart horizontally, but their vertical position remains constant. For quadratic functions, understanding horizontal stretches is particularly important, as they affect the width of the parabola but not the direction in which it opens.

Impact of Transformation on the Original Function

• Alters the graph's shape without changing the direction it opens.
• Maintains the y-intercept while stretching or compressing the x-intercepts.
• Keeps the parabola's vertex in the same vertical line, but shifts it horizontally.

Recognizing these shifts is key to understanding how changing units or variables affects the behavior of a function.

Units Conversion in Functions

Converting units in functions is an essential skill, as it allows for consistency and comparability across different measurement systems. In the exercise, speeds measured in miles per hour (mph) need to be converted into kilometers per hour (kph) to adapt the function accordingly, by recognizing that 1 mile is equivalent to 1.6 kilometers.

Unit conversion within functions does not only involve a direct multiplicative factor (like multiplying by 1.6 in linear functions) but can also lead to more complex transformations in the case of quadratic functions. When dealing with quadratic terms, as seen in the horsepower function, one must apply the unit conversion factor squared (\(1.6^2\)) to the quadratic term, but only once to the linear term.

Practical Steps for Conversion

• Identify the conversion factor between the units.
• Apply the conversion factor to each term accordingly. For quadratic terms, use the squared factor; for linear terms, use the factor as is.
• Re-calculate any constants to accommodate the new units.

The mastery of units conversion is a vital tool, enabling students to navigate through diverse scientific and mathematical problems with ease and precision.