Question

Write an equation for a function that has the given graph. Line segment connecting (-4,3) and (0,-5)

Short answer

The equation of the line segment connecting the points (-4,3) and (0,-5) is \(y = -2x - 5\)

Step-by-step solution

Determine the Slope

The slope of a line segment between two points (x1, y1) and (x2, y2) is determined using the formula: \(m = \frac{y2 - y1}{x2 - x1}\). Substituting the given points into the formula results in: \(m = \frac{-5 - 3}{0 - (-4)} = -2\)

Use the Slope and a Point to Determine the Y-Intercept

Now that we have the slope, we can use it along with one of the points to derive the equation of the line in slope-intercept form, \(y = mx + b\). We can use the point (0,-5) to do this. We substitute into this equation and solve for b: -5 = -2*(0) + b results in b = -5.

Write the Equation of the Line

With the slope (\(m=-2\)) and the y-intercept (\(b=-5\)), we can write the equation of the line as \(y = -2x - 5\)

Key concepts

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a line so that its slope and y-intercept are immediately apparent. This form is represented by the equation \( y = mx + b \), where \( m \) stands for the slope of the line and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

When constructing an equation from a graph, the slope-intercept form is particularly useful. It simplifies the process to a two-step journey: find the slope (\( m \)) and find the y-intercept (\( b \)). Given any two points on the line, you can calculate the slope, and using the slope with any one of those points, you can calculate the y-intercept. Once both \( m \) and \( b \) are found, they are plugged into the slope-intercept form, and you immediately have the equation that represents the line on the graph.

Understanding the slope-intercept form is essential in algebra and plays a significant role in various applications across mathematics, including economics and physics.

Linear Equations

Linear equations form the foundation of a vast area in algebra and represent straight-line graphs on a coordinate plane. They can be portrayed in various forms, including the slope-intercept form, the point-slope form, and the standard form, among others. Regardless of the form, a linear equation in two variables, like \( x \) and \( y \), can always be simplified to create a straight line when graphed.

The beauty of linear equations lies in their simplicity and the ease with which they can be manipulated to understand relationships between variables. For instance, if you have the equation of a line, you can figure out the coordinates of any point on that line by plugging in the value for either \( x \) or \( y \).

Moreover, linear equations are not just theoretical concepts. They are used in real-world scenarios to predict trends, analyze rates of change, and model various phenomena. The ability to write and solve linear equations is a skill that proves useful in fields ranging from engineering to social science.

Slope of a Line

The slope of a line is a measure of its steepness and direction. In mathematical terms, it is the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points on the line. The formula to calculate the slope between two points \((x1, y1)\) and \((x2, y2)\) is \( m = \frac{y2 - y1}{x2 - x1} \).

A positive slope means the line inclines upward as it moves from left to right, while a negative slope indicates the line declines. If the slope is zero, the line is horizontal, and an undefined slope (division by zero) corresponds to a vertical line.

It's important to note that two lines are parallel if they have the same slope and are perpendicular if the product of their slopes is -1. This concept of slope equips students not only with the understanding to describe linear relationships but also with the basis for exploring more complex functions and their graphs.