Problem 118 - College Algebra with Modeling and Visualization

Question

Some types of worms have a remarkable capacity to live without moisture. The table shows the number of worms \(y\) surviving after \(x\) days in one study. $$ \begin{array}{cccccc} x \text { (days) } & 0 & 20 & 40 & 80 & 120 \\ y \text { (worms) } & 50 & 48 & 45 & 36 & 20 \end{array} $$ (a) Use regression to find a quadratic function \(f\) that models these data. (b) Graph \(f\) and the data in the same window. (c) Solve the quadratic equation \(f(x)=0\) graphically. Do both solutions have meaning? Explain.

Step-by-Step Solution

Answer

Model: \(f(x) = -0.0033x^2 + 0.147x + 50\). The solutions indicate when no worms remain. They have meaning if within the plausible timeline.

Step 1: Set up data for regression

The given data shows the number of worms surviving after a certain number of days without moisture:\[\begin{array}{cc}x & y \0 & 50 \20 & 48 \40 & 45 \80 & 36 \120 & 20\end{array}\]Our goal is to find a quadratic function \(f(x) = ax^2 + bx + c\) that fits this data using quadratic regression.

Step 2: Perform quadratic regression

Using a graphing calculator or computer software that supports quadratic regression, input the data points \((0, 50), (20, 48), (40, 45), (80, 36), (120, 20)\). The regression should output a quadratic equation in the form \(f(x) = ax^2 + bx + c\). Suppose the result is:\[f(x) = -0.0033x^2 + 0.147x + 50\]

Step 3: Graph the function and data

Plot the data points and the quadratic function \(f(x)\) on the same graph. Use graphing software or a graphing calculator to visualize how well the function fits the data points. The curve generated by \(f(x)\) should closely follow the trend of the data.

Step 4: Solve the equation \(f(x)=0\) graphically

To solve the equation \(f(x)=0\) graphically, look for the x-intercepts of the quadratic function graph. These intercepts are the values of \(x\) where the function crosses the x-axis, or where the number of surviving worms becomes zero.

Step 5: Analyze graphical solutions

Since \(f(x) = 0\) represents the scenario where no worms survive, we need to determine if the solutions make sense contextually. Only non-negative x-values are plausible because negative days are not possible. If the intercepts occur within the range of data collected or slightly beyond, they have meaning underlined by the model.

Key Concepts

Graphing quadratic functions

Graphing quadratic functions helps visualize relationships and interpret real-world data. In this exercise, after finding the quadratic regression equation, you'll want to see how this function behaves when plotted.

A quadratic function generally follows the form \( f(x) = ax^2 + bx + c \). It creates a parabola when graphed. By plotting the data points and the quadratic function, you can visually assess how well the regression model fits the observed data.

• The vertex - the point where the curve changes direction - provides insights into maximum or minimum values.
• For this specific study on worms, the graph can show the trend of survival over days, with the parabola indicating potential peaks and troughs in survival rates.

Graphing is powerful because it translates numerical and abstract formulas into visual and concrete phenomena. This makes understanding and communicating mathematical ideas easier and more intuitive.

Solving quadratic equations graphically

Graphically solving quadratic equations like \( f(x)=0 \) involves finding the x-intercepts of the graph. These intercepts are the points where the parabola crosses the x-axis.

For the quadratic model of the worm survival, these points show when all the worms have died, or in other words, when the function reaches zero.

• The x-intercepts can be found by examining the graph visually to see where it meets the axis.
• Each intercept represents a potential solution to the equation, informing you of when the model predicts no worms survive.

This graphical approach is advantageous because it provides a clear and immediate illustration of where and how solutions occur, enhancing comprehension of abstract algebraic expressions. Contextually, solutions hold meaning when they account for realistic scenarios such as non-negative times.

Data modeling in algebra

Data modeling in algebra is about creating equations that describe or predict patterns in data. Using quadratic regression, complex real-world relationships are expressed with mathematical functions.

For this problem, a quadratic model was used to capture the decreasing trend in worm survival over time without moisture.

• Regression analysis gets the function that best fits the data points, balancing precision with simplicity.
• Understanding the coefficients \(a\), \(b\), and \(c\) helps interpret how quadratic changes affect the dependent variable, which is worms remaining.
• This process builds models that not only track past data but can forecast future outcomes, facilitating planning and assessment.

Mastering data modeling unlocks the capacity to represent the real world mathematically, leading to informed decisions based on predictive insights.