Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. The heart rate of an athlete while weight training is recorded for 4 minutes. The table lists the heart rate after \(x\) minutes. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Time (min) } & 0 & 1 & 2 & 3 & 4 \\ \hline \begin{array}{l} \text { Heart rate } \\ \text { (bpm) } \end{array} & 84 & 111 & 120 & 110 & 85 \\ \hline \end{array}$$ (a) Explain why the data are not linear. (b) Find a quadratic function \(f\) that models the data. Use \((2,120)\) as the vertex of the parabola. (c) What is the domain of the function?

Step-by-Step Solution

Answer

(a) Data are not linear as changes aren't constant. (b) Function: \( f(x) = -9(x - 2)^2 + 120 \). (c) Domain: [0, 4].

Step 1: Verify Linearity

To determine linearity, the change in heart rates per minute should be constant. Calculate the differences between consecutive heart rates: - From 0 to 1 min: 111 - 84 = 27 - From 1 to 2 min: 120 - 111 = 9 - From 2 to 3 min: 110 - 120 = -10 - From 3 to 4 min: 85 - 110 = -25 The changes are not constant, so the data are not linear.

Step 2: Identify Form of Quadratic Function

A quadratic function is generally in the form \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Here, the vertex is \((2, 120)\), so the function is \( f(x) = a(x - 2)^2 + 120 \).

Step 3: Determine Coefficient 'a'

Use another data point to solve for \( a \). Let's use (0, 84):Substitute into the equation: \( 84 = a(0 - 2)^2 + 120 \)Solve for \( a \):\( 84 = 4a + 120 \)\( 4a = 84 - 120 \)\( 4a = -36 \)\( a = -9 \)Thus, the quadratic function is \( f(x) = -9(x - 2)^2 + 120 \).

Step 4: Determine the Domain of the Quadratic Function

Since the context is time from 0 to 4 minutes, the domain of the function \( f \) is \( 0 \leq x \leq 4 \).

Key Concepts

Non-linear Data

When analyzing a set of data points, one key concept is to determine whether the data exhibits linear or non-linear characteristics. For data to be considered linear, the rate of change between each consecutive data point must remain constant. In our example, we are observing an athlete's heart rate over four minutes. Looking at the changes in the heart rate:

• From 0 to 1 minute, the heart rate increases by 27 bpm (beats per minute).
• From 1 to 2 minutes, the increase is 9 bpm.
• From 2 to 3 minutes, there is a decline of 10 bpm.
• From 3 to 4 minutes, the rate drops further by 25 bpm.

These changes differ over time, indicating that the data is non-linear. Such non-linear patterns are typical in many real-world scenarios where changes occur irregularly, like heart rates responding to physical activity.

Vertex Form of a Quadratic

Quadratic functions can be presented in various forms, and the vertex form is particularly useful when the vertex of the curve is needed. The vertex form of a quadratic function is given as:\[ f(x) = a(x - h)^2 + k \]Here, \((h, k)\) represents the vertex of the parabola. In our exercise, we use \((2, 120)\) as the vertex, placing the values directly into the function:\[ f(x) = a(x - 2)^2 + 120 \]To identify the parameter \(a\), which dictates the parabola's concavity and stretch, another point is required. Using the point \((0, 84)\), we substitute into the equation and solve:

• Substitute \(0\) for \(x\) and \(84\) for \(f(x)\): \(84 = a(0 - 2)^2 + 120\).
• Solve for \(a\) which results in \(a = -9\).

Our final quadratic function becomes:\[ f(x) = -9(x - 2)^2 + 120 \]This specific form allows us to directly see the highest point of the heart rate (120 bpm at 2 minutes) and establishes how the rate changes over time.

Domain of a Function

The domain of a function refers to all the possible values that the independent variable can take. In our context, since the heart rate data was recorded over a time span of 0 to 4 minutes, the domain is restricted to this interval.

• In mathematical terms, we express this domain using the inequality: \(0 \leq x \leq 4\).
• Here, \(x\) represents time in minutes, indicating that any calculations or predictions using the quadratic function should fall within this timeframe.

Domains are essential in ensuring that the function is only applied to relevant data points, maintaining the practical limits of the scenario. They allow us to apply the quadratic model only to the time period in which the athlete's heart rate was observed.