Problem 89 - Precalculus: Functions and Graphs

Question

Choose the equation that best describes the table of data. (Hint: Make assignments to \(\mathbf{Y}_{\mathbf{r}}-\mathbf{Y}_{\mathbf{4}}\) and examine a table of their values.) $$\begin{array}{|c|c|}\hline x & y \\\\\hline 1 & -9 \\\\\hline 2 & -4 \\\\\hline 3 & 11 \\\\\hline 4 & 42 \\\\\hline 5 & 95 \\\\\hline\end{array}$$ (1) \(y=13 x-22\) (2) \(y=x^{2}-2 x-8\) (3) \(y=4 \sqrt{x}-13\) (4) \(y=x^{3}-x^{2}+x-10\)

Step-by-Step Solution

Answer

The equation is \(y = x^3 - x^2 + x - 10\).

Step 1: Calculate Differences in y Values

Begin by calculating the differences in the given y-values. Determine a pattern by looking at how y-values change as x increases.\[\begin{align*}Y_2 - Y_1 & = -4 - (-9) = 5 \Y_3 - Y_2 & = 11 - (-4) = 15 \Y_4 - Y_3 & = 42 - 11 = 31 \Y_5 - Y_4 & = 95 - 42 = 53 \\end{align*}\]These differences (5, 15, 31, 53) indicate that the change is not constant, suggesting a non-linear function.

Step 2: Identify a Pattern in Differences

Look for a pattern in the differences calculated previously. Notice that these differences themselves are increasing: 5, 15, 31, 53. Compute the differences of these differences.\[\begin{align*}15 - 5 &= 10 \31 - 15 &= 16 \53 - 31 &= 22 \\end{align*}\]These second differences (10, 16, 22) suggest a quadratic or higher-degree relationship.

Step 3: Test Possible Equations

Evaluate the provided equations to match the table data. We suspect a higher degree polynomial might fit due to the differences.**Option 1**: \(y = 13x - 22\)\[\begin{array}{c c}x & y = 13x - 22 \1 & -9 \2 & 4 \3 & 17 \4 & 30 \5 & 43 \\end{array}\]This does not match the table.**Option 2**: \(y = x^2 - 2x - 8\)\[\begin{array}{c c}x & y = x^2 - 2x - 8 \1 & -9 \2 & -2 \3 & 1 \4 & 12 \5 & 27 \\end{array}\]This does not match the table.**Option 3**: \(y = 4\sqrt{x} - 13\)This option is unlikely based on square roots.**Option 4**: \(y = x^3 - x^2 + x - 10\)\[\begin{array}{c c}x & y = x^3 - x^2 + x - 10 \1 & -9 \2 & -4 \3 & 11 \4 & 42 \5 & 95 \\end{array}\]This matches the table perfectly.

Key Concepts

Differences in sequences

When analyzing a sequence of numbers or a set of data points, one practical approach is to investigate the differences between successive terms. This concept, often called 'differences in sequences,' helps to uncover patterns and characteristics of the data. In the case of the problem, we have a table with x and corresponding y values. By calculating the difference between successive y-values, we can determine if the change between terms is constant or varies.

Let's illustrate how this process helps identify the type of function representing the data:

• Calculate the first differences, which are the changes between consecutive y-values: \(Y_2 - Y_1, Y_3 - Y_2, ext{etc.}\).
• If the first differences are not constant, it suggests a non-linear function.
• Next, compute the second differences; these are the differences between the first differences.
  If they are constant, this might indicate a quadratic relationship.

In our given problem, the differences increase (5, 15, 31, 53), and further examination shows that even the second differences (10, 16, 22) are not consistent. This means the function is not linear or quadratic, suggesting the presence of a more complex relationship, likely a higher-degree polynomial.

Non-linear functions

Non-linear functions are functions in which the graph does not form a straight line. These differ from linear functions, and recognizing them is crucial in interpreting various real-world phenomena. They can involve powers of variables other than 1, roots, or even graph features like curves.

In problems involving non-linear functions, identifying their form early can simplify the task of finding the right equation for a dataset. A dataset where changes between values are not uniform is an indication of non-linearity. In the workout problem, the drastic variation in y-values indicates non-linearity. The first set of differences verified that the sequence of y-values doesn't align with a linear relationship.

• Determine if they fit a well-known function type, such as quadratic or cubic, or another polynomial type.
• Utilize the fact that quadratic functions show constant second differences.
  If these too vary, the function may be cubic or involve even higher powers.

Recognizing the non-linear nature of a function is key to finding its equation. The objective is to identify equations precisely matching the values in the problem's dataset.

Higher-degree polynomials

Higher-degree polynomials are polynomial functions with a degree of three or more. The degree of a polynomial is the highest power of the variable in the function. This exercise involves identifying whether a higher-degree polynomial, specifically a third-degree (cubic) polynomial, fits the data.

In math problems, higher-degree polynomials can represent more complex relationships that simple linear or quadratic models cannot capture. As seen in this task, the variance in differences between y-values indicates it's not merely quadratic. Testing with different possible equations might indicate that our initial assumption of the function's degree was incorrect.

• The polynomial degree provides key insights into how rapidly values can change.
• Cubic functions specifically show changing first and second differences,
  often seen in fast-growing datasets.
• Testing provided equations like a cubic polynomial and refining it ensures it matches exactly with the observed dataset.

In conclusion, the solution showed that a cubic polynomial (third-degree) accurately matched the dataset. Therefore, understanding the function type deeply impacted by the degree of the polynomial is crucial in achieving an accurate equation to describe data tables like the one given.