Problem 97 - College Algebra

Question

For the following exercises, use a calculator or graphing technology to complete the task. If \(f\) is a linear function, \(f(0.1)=11.5,\) and \(f(0.4)=-5.9,\) fi \(\mathrm{d}\) an equation for the function.

Step-by-Step Solution

Answer

The equation of the linear function is \(f(x) = -58x + 17.3\).

Step 1: Identify Key Information

We are given two points on the line: \((0.1, 11.5)\) and \((0.4, -5.9)\). These points will help us determine the equation of the linear function.

Step 2: Calculate the Slope

Use the formula for the slope, \(m\), of a line passing through two points: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points, \(m = \frac{-5.9 - 11.5}{0.4 - 0.1} = \frac{-17.4}{0.3} = -58\).

Step 3: Write the Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\). We know \(m = -58\) from the previous step.

Step 4: Calculate the Y-Intercept

Choose one of the given points to solve for \(b\). Using the point \((0.1, 11.5)\), substitute into the equation: \(11.5 = -58(0.1) + b\). This simplifies to \(11.5 = -5.8 + b\), so \(b = 17.3\).

Step 5: Write the Final Equation

Substituting the slope \(-58\) and the y-intercept \(17.3\) into the slope-intercept form, we get \(f(x) = -58x + 17.3\).

Key Concepts

Slope Calculation

The slope of a line is a key component in understanding linear functions. It tells us how steep the line is and the direction it goes. For any two points on a line, the slope is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here's what this means:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the line.
• \( y_2 - y_1 \) is the change in the y-values (vertical change).
• \( x_2 - x_1 \) is the change in the x-values (horizontal change).
• \( m \) is the slope of the line.

In our example, the points \((0.1, 11.5)\) and \((0.4, -5.9)\) are used. Substituting into the formula, we get the slope as \( m = \frac{-5.9 - 11.5}{0.4 - 0.1} = -58 \). This negative slope indicates that the line is decreasing (or falling) as it moves from left to right.

Y-Intercept

The y-intercept of a line is where it crosses the y-axis. This occurs when the x-value is zero, which means the line meets the y-axis at the point \( (0, b) \), where \( b \) is the y-intercept.To find the y-intercept, we use the slope-intercept equation \( y = mx + b \) and substitute one of the known points on the line. Let's use the point \((0.1, 11.5)\):

• Substitute \( x = 0.1 \) and \( y = 11.5 \) into the equation.
• Use the previously calculated slope \( m = -58 \).
• The equation becomes \( 11.5 = -58(0.1) + b \).

Solving for \( b \), you add the product of the slope and x-value to \( 11.5 \), giving \( b = 17.3 \). Thus, the y-intercept is \( 17.3 \), and the line meets the y-axis at the point \( (0, 17.3) \).

Slope-Intercept Form

The slope-intercept form is a straightforward representation of a linear equation: \( y = mx + b \). This form makes it easy to graph a linear function and understand its properties.

• \( m \) represents the slope – it shows the rate of change and direction of the line.
• \( b \) is the y-intercept – the point where the line crosses the y-axis.
• Knowing these two values allows you to quickly sketch the line.

With our example, substituting the slope \( m = -58 \) and y-intercept \( b = 17.3 \) gives us the linear equation \( f(x) = -58x + 17.3 \). This formula encapsulates all the essential characteristics of the line, making it a useful tool for both analysis and graphing.