Problem 91 - College Algebra

Question

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 5 & 10 & 15 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & -5 & 20 & 45 & 70 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The table represents a linear function: \( f(x) = 5x - 5 \).

Step 1: Identify Linear Relationship

To determine if a table represents a linear function, check if the difference between consecutive \( f(x) \) values (outputs) is constant. Calculate the differences for each successive pair of outputs.

Step 2: Calculate Differences

Calculate the differences between \( f(x) \) values: \( f(5) - f(0) = 20 - (-5) = 25 \), \( f(10) - f(5) = 45 - 20 = 25 \), and \( f(15) - f(10) = 70 - 45 = 25 \). All differences are consistent, which confirms a linear relationship.

Step 3: Determine the Slope

The constant difference found in Step 2 is the change in \( f(x) \) per change in \( x \). The formula for slope is \( m = \frac{\Delta f(x)}{\Delta x} \). Thus, the slope is \( m = \frac{25}{5} = 5 \).

Step 4: Find the Y-intercept

Use a known point to find the y-intercept \( b \) in the equation \( f(x) = mx + b \). Using point \((0, -5)\): \( -5 = 5 \times 0 + b \). Thus, \( b = -5 \).

Step 5: Write the Equation

Now that we have the slope \( m = 5 \) and y-intercept \( b = -5 \), write the equation of the line: \( f(x) = 5x - 5 \).

Key Concepts

Slope Determination

The concept of slope is central when working with linear functions. Slope tells us how steep a line is and the direction in which it inclines or declines. To determine the slope of a linear function from a table like the one we have, you need to calculate the change in the function values, \(f(x)\), over the change in \(x\) values. This change is typically represented by delta, \(\Delta\).

• The difference in \(f(x)\) values, known as \(\Delta f(x)\), is calculated by taking the difference between consecutive function values.
• The difference in \(x\) values, known as \(\Delta x\), is generally the difference between consecutive \(x\) values, which is usually constant in a table.

In our example, each step in \(f(x)\) is consistently 25, and each \(x\) value increases by 5, leading to a slope \(m\) computed as \(m = \frac{25}{5} = 5\).This slope tells us that for every 5-unit increase in \(x\), \(f(x)\) increases by 25.

Y-intercept

The y-intercept is a key parameter in the equation of a line. It indicates the point at which the line crosses the y-axis (when \(x = 0\)). Finding the y-intercept involves using the equation of a line \(f(x) = mx + b\), where \(b\) denotes the y-intercept.To find \(b\), select a point from the data where \(x = 0\). In this exercise, the point is \((0, -5)\). You've already determined the slope \(m = 5\). Substituting into the equation, you have \(-5 = 5 \times 0 + b\). Solving for \(b\), we find \(b = -5\).This means the line crosses the y-axis at \(y = -5\), giving essential insight into the starting point or inherent value of the function when \(x\) is zero.

Linear Equations

Linear equations are foundational in mathematics and essential in analyzing relationships between variables. They represent a constant rate of change, displayed graphically as a straight line. The general form of a linear equation is \(f(x) = mx + b\),

• where \(m\) is the slope of the line, indicating the steepness and direction of the linear relationship.
• The \(b\) is the y-intercept, revealing where the line touches the y-axis.

In this particular exercise, we determined a linear equation from the table data: \(f(x) = 5x - 5\). This indicates that the function increases by 5 units for every unit increase in \(x\), and it starts at \(-5\) on the \(y\)-axis. Understanding the structure of linear equations is crucial, as it ties the visual representation of a line to its algebraic expression, aiding in predicting values and interpreting data effectively.