Problem 8 - Algebra Form and Function

Question

Could the table represent the values of a linear function? Give a formula if it could. $$ \begin{array}{c|c|c|c|c} \hline x & 0 & 2 & 10 & 20 \\ \hline y & 50 & 58 & 90 & 130 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

If yes, what is the formula for the linear function? Answer: Yes, the table represents the values of a linear function. The formula for the linear function is y = 4x + 50.

Step 1: Calculate the differences in y values for corresponding x values

We will calculate the differences in y values (Δy) for corresponding differences in x values (Δx) for each consecutive pair of values in the table. Δy1: $$ 58-50 = 8 $$ Δy2: $$ 90-58 = 32 $$ Δy3: $$ 130-90 = 40 $$ Now, we'll also find the differences in x values (Δx) for these pairs: Δx1: $$ 2-0 = 2 $$ Δx2: $$ 10-2 = 8 $$ Δx3: $$ 20-10 = 10 $$

Step 2: Check if the differences in y values are proportional to the differences in x values

Now, let's compare the ratios of Δy to Δx for each pair. If these ratios are the same for all pairs, then the table represents the values of a linear function. Ratio1: $$ \frac{Δy1}{Δx1} = \frac{8}{2} = 4 $$ Ratio2: $$ \frac{Δy2}{Δx2} = \frac{32}{8} = 4 $$ Ratio3: $$ \frac{Δy3}{Δx3} = \frac{40}{10} = 4 $$ As we can see, the ratios for all pairs are the same, which means that the table does represent the values of a linear function.

Step 3: Find the formula for the linear function

Now that we have verified that the table represents the values of a linear function, we'll find the formula for that function. A linear function is represented by the equation: $$ y = mx + b $$ Where m is the slope, x is the independent variable, and b is the y-intercept. We already know the slope (m), which is the ratio that we found in Step 2: $$ m = 4 $$ Using the first point in the table (x = 0 and y = 50), we can now find the y-intercept (b): $$ 50 = 4(0) + b $$ Solving for b, we get: $$ b = 50 $$ Now, we have all the information needed to find the formula for the linear function: $$ y = 4x + 50 $$ So, the table does represent the values of a linear function, and the formula for that function is: $$ y = 4x + 50 $$

Key Concepts

Slope-Intercept Form

Linear functions are often expressed using the slope-intercept form, which is written as \( y = mx + b \). This form is a straightforward way to understand linear relationships. In this equation:

• \( y \) is the dependent variable.
• \( x \) is the independent variable or input value.
• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the value of \( y \) when \( x = 0 \).

The slope \( m \) tells us how much \( y \) changes for a change in \( x \). In other words, it's the rise over run between two points. The y-intercept \( b \) shows where the line crosses the y-axis. It's a critical starting point when graphing a line.

In the given exercise, the linear equation derived is \( y = 4x + 50 \). Here, \( m = 4 \) indicates that for every unit increase in \( x \), \( y \) increases by 4. The y-intercept \( b = 50 \) tells us that when \( x = 0 \), \( y \) equals 50.

Mathematical Ratios

Mathematical ratios play a crucial role in determining if a set of data represents a linear function. The key idea is to check if changes in \( y \) are proportional to the changes in \( x \). This is done by computing the ratio of change in \( y \) to change in \( x \) between each pair of points.

• Step 1: Calculate \( Δy \), the difference in \( y \) values.
• Step 2: Calculate \( Δx \), the difference in \( x \) values.
• Step 3: Find \( \frac{Δy}{Δx} \), the ratio for each point pair.

In the exercise, the ratios \( \frac{Δy}{Δx} \) for all data pairs were calculated as follows: \( \frac{8}{2} = 4 \), \( \frac{32}{8} = 4 \), and \( \frac{40}{10} = 4 \). Since these ratios remained constant, it verified the data represents a linear function. This constant ratio equates to the slope of the linear function, confirming uniform growth or decline across the dataset.

Function Table Analysis

Function table analysis is a valuable method for understanding the behavior of functions using discrete data points. By analyzing a table of values, you can determine whether the relationship between variables is linear. Here are the steps needed in this analysis:

• Identify pairs of \( x \) and \( y \) values.
• Examine the changes in \( y \) values compared to the changes in \( x \) values.
• Check if these changes are consistent across the table.

In the provided table, we see it contains values for \( x \) and \( y \) which, upon investigation, show that differences in \( y \) are consistently proportional to differences in \( x \). When performing function table analysis, noticing these consistent steps helps to find the formula directly using the linear function equation \( y = mx + b \). This confirms that the table supports a linear function and aids in deriving its equation through observed data.