Question

Find the slope of the graph of the linear function \(f\). $$ f(6)=-1, f(3)=8 $$

Short answer

The slope of the graph of the function \(f\) is -3

Step-by-step solution

Identify the two points

From the given function values, you can determine two points are on the graph of the function: \((3, 8)\) and \((6, -1)\).

Apply the slope formula

Now, apply the slope formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\) using the coordinates of the two points. Substituting the given values, the equation becomes \(m = \frac{{(-1) - 8}}{{6 - 3}}\).

Simplify the equation

Simplify the equation from Step 2 to get the slope \(m\). Calculation: \(m = \frac{{-9}}{{3}} = -3\)

Key concepts

Slope Formula

Understanding the slope formula is crucial when it comes to analyzing the steepness or direction of a line on a graph. The formula is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( m \) represents the slope, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two distinct points on the line.

It's important to note that the slope reflects how many units the line goes up (or down) for each unit it moves to the right. A positive slope indicates an upward trend, while a negative slope shows a downward trend. A slope of zero suggests a horizontal line, indicating no change, and if the slope is undefined, this represents a vertical line where there is no horizontal change.

When using the slope formula, you're essentially calculating the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.

Linear Function

A linear function is one of the simplest forms of functions in algebra and is defined by an equation of the first degree, meaning it has no exponents greater than one. Such functions are graphically represented by straight lines. The general form of a linear function is \( f(x) = mx + b \), where \( m \) is the slope and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

The behavior and direction of the line are determined by the slope, while its position on the graph is influenced by the y-intercept. Understanding the role of these components allows you to sketch the function's graph accurately and understand the relationship between variables in real-world problems.

Slope Calculation

To effectively calculate the slope of a linear function, you begin by identifying two points on the graph of the function, as done in the step-by-step solution of the textbook problem. Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), you can calculate the slope by subtracting the y-coordinate of the first point from the y-coordinate of the second point, and placing this over the subtraction of the x-coordinate of the first point from the x-coordinate of the second point.

For example, with our given points \( (3, 8) \) and \( (6, -1) \), the calculation becomes \( m = \frac{(-1) - 8}{6 - 3} = \frac{-9}{3} = -3 \). The calculated slope, in this case, is \( -3 \), indicating that the line descends three units vertically for every one unit it moves horizontally.

Coordinate Points

Coordinate points are used to determine the position of a point in a two-dimensional space, such as a graph. The format for these points is \( (x, y) \), where \( x \) is the value on the horizontal axis (abscissa) and \( y \) is the value on the vertical axis (ordinate).

To work with linear functions and calculate their slope, you need at least two coordinate points. It’s essential to carefully choose points that accurately represent the function. In homework problems, such points are often provided, or they can be determined from the function itself, as seen when the function values \( f(6) \) and \( f(3) \) were given. These functions help us pinpoint the exact locations of \( (6, -1) \) and \( (3, 8) \) on the graph, providing clear reference points for further calculations.