Problem 41 - Algebra 2

Question

Find the slope and \(y\) -intercept of the graph of each function. $$ y=2 x-6 $$

Step-by-Step Solution

Answer

The slope of the graph is 2 and the y-intercept is \((0, -6)\).

Step 1: Identify the Slope and Y-intercept

The equation is already in slope-intercept form, which is given by \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. In the equation \(y = 2x - 6\), the slope \(m\) is the coefficient of \(x\), which is 2, and the y-intercept \(b\) is the constant term, which is -6.

Step 2: Write Down the Slope and Y-intercept

The slope of the graph is 2, and the y-intercept is the point on the y-axis where the line crosses it, which is at \(y = -6\). Therefore, the y-intercept as a coordinate is \((0, -6)\).

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and is a vital concept in understanding linear relationships. In mathematical terms, the slope is the ratio of the rise (the change in the vertical direction) to the run (the change in the horizontal direction). The formula to calculate the slope (\(m\)) between two points \(x_1, y_1\) and \(x_2, y_2\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In the context of the equation \(y = 2x - 6\), the slope is simply the coefficient of \(x\), which is 2. This means for every one unit you move to the right (along the x-axis), the value of \(y\) increases by two units. If the slope is positive, like in this case, the line inclines upwards as you move from left to right. Conversely, a negative slope would mean the line inclines downwards as you move in the same direction.
Understanding the concept of slope allows you to predict and understand how changes in one variable affect another. For example, if you think of \(x\) as time and \(y\) as distance traveled, a slope of 2 implies that the distance increases by 2 units for every unit of time—that is, the speed is constant at 2 units per time period.

Y-intercept

In algebra, the y-intercept of a line is the point where the line crosses the y-axis. It is important because it gives us a starting point of the line when \(x=0\). The y-intercept is always expressed as a coordinate with an \(x\) value of 0. For the equation \(y = 2x - 6\), finding the \(y\)-intercept is straightforward since the equation is in the slope-intercept form \(y=mx+b\), where \(b\) is the y-intercept. Here, \(b=-6\), indicating that the line will cross the y-axis at \(y=-6\).

As such, the coordinate of the y-intercept is \(\left(0, -6\right)\). Visualizing this on a graph, you can imagine plotting the point (0, -6) on the y-axis and seeing the line extend in both directions with a slope that makes it rise by 2 units for every 1 unit it moves to the right, forming the character of the line's trajectory.

Linear Equations

Linear equations form the foundation for a large area of algebra and are characterized by variables raised to the first power and graphed as straight lines. A common form of a linear equation is the slope-intercept form, expressed as \(y=mx+b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept.

Linear equations describe a constant rate of change between two variables. They are used to model real-world phenomena such as speed, economic forecasts, and scientific relationships. Students can utilize these equations to solve problems that require finding how quantities relate to each other.

Calculating with Linear Equations

To solve linear equations, one can manipulate the equation using arithmetic operations to isolate the variable of interest, typically \(y\). In the exercise \(y = 2x - 6\), the equation already tells you directly how \(y\) will change with every change in \(x\) because it is in slope-intercept form, making it easier to graph and understand. This form is particularly useful when analyzing data, predicting outcomes, or solving optimization problems within linear constraints.