91Ó°ÊÓ

Which of the following tables could represent a linear function? For each that could be linear, find a linear equation models the data. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{g}(\boldsymbol{x}) \\ \hline 0 & 5 \\ \hline 5 & -10 \\ \hline 10 & -25 \\ \hline 15 & -40 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{h}(\boldsymbol{x}) \\ \hline 0 & 5 \\ \hline 5 & 30 \\ \hline 10 & 105 \\ \hline 15 & 230 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\ \hline 0 & -5 \\ \hline 5 & 20 \\ \hline 10 & 45 \\ \hline 15 & 70 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{k}(\boldsymbol{x}) \\ \hline 5 & 13 \\ \hline 10 & 28 \\ \hline 20 & 58 \\ \hline 25 & 73 \\ \hline \end{array} $$

Step by step solution

01

Check for Constant Differences in Table 1 (g(x))

Calculate the differences between consecutive values of \( g(x) \):- From 5 to -10: \(-10 - 5 = -15\)- From -10 to -25: \(-25 - (-10) = -15\)- From -25 to -40: \(-40 - (-25) = -15\)Since the differences are constant, **\( g(x) \) could represent a linear function**.

02

Develop Linear Equation for g(x)

Using the formula for a linear function, \( y = mx + b \):- Slope \( m = \frac{-15}{5} = -3\)- Using point \((0, 5)\), the equation is \( g(x) = -3x + 5 \).

03

Check for Constant Differences in Table 2 (h(x))

Calculate the differences between consecutive values of \( h(x) \):- From 5 to 30: \(30 - 5 = 25\)- From 30 to 105: \(105 - 30 = 75\)- From 105 to 230: \(230 - 105 = 125\)Since the differences are not constant, **\( h(x) \) does not represent a linear function**.

04

Check for Constant Differences in Table 3 (f(x))

Calculate the differences between consecutive values of \( f(x) \):- From -5 to 20: \(20 - (-5) = 25\)- From 20 to 45: \(45 - 20 = 25\)- From 45 to 70: \(70 - 45 = 25\)The differences are constant, so **\( f(x) \) could represent a linear function**.

05

Develop Linear Equation for f(x)

Using the point-slope form, the slope \( m = \frac{25}{5} = 5 \). Using the point \((0, -5)\), the equation is \( f(x) = 5x - 5 \).

06

Check for Constant Differences in Table 4 (k(x))

Calculate the differences between consecutive values of \( k(x) \):- From 13 to 28: \(28 - 13 = 15\)- From 28 to 58: \(58 - 28 = 30\)- From 58 to 73: \(73 - 58 = 15\)Since the differences are not constant, **\( k(x) \) does not represent a linear function**.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference Method

When working with data tables, students often need to determine whether a set of points can be represented by a linear function. One common approach for this is the **difference method**. This method involves checking for constant differences between consecutive outputs.
To use the difference method effectively:

• Examine the increments between consecutive x-values. They should be consistent for a fair comparison of y-values.
• Calculate the differences between each pair of successive y-values. Simply subtract the previous y-value from the current one.

Consider Table 1, representing function values for \( g(x) \):

• The x-values increase by a constant 5 units. That is a good start.
• The y-differences: \(-15, -15, -15\). They are constant, indicating a linear relationship.

This means that the differences are consistently spaced, confirming that the function could be linear. This method shines because it provides a quick validation without deep calculations.

Slope-Intercept Form

Once you establish that the relationship between the x and y values is linear, the next step is to express this relationship in the **slope-intercept form**. This is a staple in linear equations, given by \( y = mx + b \).
Here's a breakdown of this form:

• \( m \) represents the slope. It is derived from those consistent differences you previously calculated. It's a measure of the steepness of the line.
• \( b \) is the y-intercept. It's the value of \( y \) when \( x \) is zero. It's where the line crosses the y-axis.

To craft an equation from Table 1 (\( g(x) \)):

• The slope \( m \) is \(-3\), calculated based on y-differences (\(-15/5 = -3\)).
• The y-intercept \( b \) is \(5\), from when \( x \) equals zero.
• The resulting equation is \( g(x) = -3x + 5 \).

Using slope-intercept form makes expressing linear relationships clear and concise.

Linear Equation Development

Developing a robust **linear equation** is essential once you've determined a dataset is linear. The process stems from converting raw data observations into a mathematical representation using steps like the difference method and applying the slope-intercept form.
Here’s a step-by-step approach you can follow:

• Identify if the relationship is linear. Use methods like the difference method.
• Calculate the slope \( m \). Ensure to use consistent x-intervals, as seen in \( g(x) \) and \( f(x) \) where slopes were calculated as \(-3\) and \(5\), respectively.
• Determine the y-intercept \( b \), which is usually derived when \( x = 0 \).
• Combine these to write the equation in \( y = mx + b \) format.

This development process simplifies the understanding of linear theories while allowing easy predictions about future values. The ability to develop these equations is a gateway to exploring and understanding the behavior of lines in the coordinate system.