Problem 132 - A Graphical Approach to College Algebra

Question

Solve each problem. The table shows a person's heart rate during the first 4 minutes after exercise has stopped. $$\begin{array}{|l|c|c|c|}\hline \text { Time (min) } & 0 & 2 & 4 \\\\\hline \text { Heart rate (bpm) } & 154 & 106 & 90\end{array}$$ (a) Find a formula \(f(x)=a(x-h)^{2}+k\) that models the data, where \(x\) represents time and \(0 \leq x \leq 4\) (b) Evaluate \(f(1)\) and interpret the result. (c) Estimate the times when the heart rate was from 115 to 125 beats per minute.

Step-by-Step Solution

Answer

(a) \(f(x) = 12(x-2)^2 + 106\). (b) \(f(1) = 118\) bpm. (c) Heart rate was 115-125 bpm from 0.742 to 3.258 minutes.

Step 1: Understanding Problem (a)

We need to find a quadratic formula in the form \(f(x) = a(x-h)^{2}+k\) that fits the data points provided in the table: (0, 154), (2, 106), and (4, 90). The challenge is to determine the values of \(a\), \(h\), and \(k\).

Step 2: Selecting the Vertex

The vertex form of a quadratic function is given by \(f(x) = a(x-h)^2 + k\), where \((h,k)\) is the vertex of the parabola. Since the data shows decreasing values of heart rate, we can choose the vertex as the midpoint of the overall time range to simplify calculations, i.e., \(h = 2\).

Step 3: Identifying the Value of k

The vertex \((h,k)\) is an actual point on the curve, thus \(f(2) = 106\), indicating that \(k = 106\).

Step 4: Solving for a

Using the fact that the point (0, 154) lies on the curve, we substitute into the formula: \ \(154 = a(0-2)^2 + 106\). Solve for \(a\): \ \(154 = 4a + 106\) \ \(48 = 4a\) \ \(a = 12\).

Step 5: Formulating the Equation

Now, substitute \(a\), \(h\), and \(k\) into the formula: \(f(x) = 12(x-2)^2 + 106\). This is the quadratic equation modeling the heart rate data.

Step 6: Evaluating f(1) for Part (b)

Substitute \(x = 1\) into the equation: \ \(f(1) = 12(1-2)^2 + 106\) \ \(f(1) = 12(1)^2 + 106\) \ \(f(1) = 12 + 106 = 118\). The result indicates that, at 1 minute after stopping exercise, the heart rate is approximately 118 bpm.

Step 7: Estimating Times for Part (c)

We need to find the times when the heart rate was between 115 bpm and 125 bpm, i.e., solve: \ \(115 \leq 12(x-2)^2 + 106 \leq 125\). 1. Start with \(12(x-2)^2 + 106 = 115\), giving: \ \(12(x-2)^2 = 9\) \ \((x-2)^2 = \frac{3}{4}\) \ \(x-2 = \pm\sqrt{0.75}\) \ \(x \approx 2 \pm 0.866\). Therefore, \(x \approx 2.866 \text{ and } 1.134\).2. For \(12(x-2)^2 + 106 = 125\), solve: \ \(12(x-2)^2 = 19\) \ \((x-2)^2 = \frac{19}{12}\) \ \(x-2 = \pm\sqrt{1.583}\). Hence, \(x \approx 2 \pm 1.258\). \ This gives \(x \approx 3.258 \text{ and } 0.742\).The heart rate was between 115 and 125 bpm at approximately 0.742 to 3.258 minutes.

Key Concepts

Vertex Form

When we talk about the vertex form of a quadratic function, we are discussing a specific way to express a quadratic equation. This form is particularly helpful in identifying key characteristics of the parabola, including its direction and vertex. The vertex form is given by:\[ f(x) = a(x-h)^2 + k \]Here, \(a\) determines how "wide" or "narrow" the parabola is and if it opens upwards or downwards. The values \((h, k)\) tell us the coordinates of the vertex of the parabola. The vertex is the "tip" or the highest or lowest point, depending on the direction in which the parabola opens. If the parabola opens upward (\(a > 0\)), the vertex is the lowest point, and if it opens downward (\(a < 0\)), the vertex is the highest point.

Using the vertex form can make it easier to graph a quadratic function since you can directly "read off" the vertex. For practical tasks like scientific data modeling or even heart rate analysis, knowing the vertex provides insights like the maximum or minimum value that the model predicts at a specific time.

Heart Rate Analysis

In the context of heart rate analysis using quadratic functions, we utilize the mathematical model to interpret and predict how heart rates change over time. Just like how in the given exercise, a set of data points representing heart rates at different times are modeled using the vertex form of a quadratic function. This technique allows us to find when the heart rate reaches certain values and helps provide a continuous model of change.

Key points to understand in heart rate analysis through mathematical modeling include:

• Recognizing patterns: A quadratic model can highlight trends in how heart rate decreases after exercise.
• Predicting future values: By using the function, you can calculate the heart rate at non-measured times.
• Analyzing specific intervals: It can be used to estimate the time when the heart rate was within a certain range, like between 115 and 125 bpm in the exercise.

By substituting different time values \(x\) into the quadratic model, predictions can be made about heart rate, making it a powerful tool for physiological studies.

Parabola Vertex

The parabola vertex is a critical point on a quadratic graph, usually representing either the maximum or minimum point in the data range (depending on whether the parabola opens upwards or downwards). Identifying the vertex is essential for understanding the turning point of the data being modeled.

For the provided exercise, the vertex was chosen based on the midpoint of the collected data because it is logical to assume that the central tendency represents a realistic point in time for analysis. In the vertex form \((h, k)\), \(h\) gives the time at which the heart rate reaches this point, while \(k\) gives the corresponding heart rate.

Understanding the vertex has practical implications:

• It can indicate equilibrium points, such as when the rate of heart rate change after stopping exercise slows down.
• In scientific contexts, it might show the point of lowest/hardest action in physical tasks.
• Helps to understand and communicate data effectively, providing a clear visual representation of the heart rate over time.

Therefore, mastering the concept of a parabola's vertex is crucial for efficiently interpreting the results of data fitted by a quadratic model like heart rates in post-exercise scenarios.