For each line, (a) find the slope and (b) sketch the graph. y = 2x - 4

Verified step by step guidance

1. Identify the equation of the line given: \(y = 2x - 4\).
2. Recall that the equation is in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
3. Compare the given equation to the slope-intercept form to find the slope \(m\). Here, \(m = 2\).
4. To sketch the graph, start by plotting the y-intercept point \((0, -4)\) on the coordinate plane.
5. From the y-intercept, use the slope \(m = 2\) (which means rise over run = 2/1) to find another point by moving up 2 units and right 1 unit, then draw the line through these points.

Key Concepts

Slope of a Line

The slope measures the steepness and direction of a line, calculated as the ratio of the change in y to the change in x (rise over run). For a line in the form y = mx + b, the slope is the coefficient m. In y = 2x - 4, the slope is 2, indicating the line rises 2 units vertically for every 1 unit it moves horizontally.

Slope-Intercept Form of a Line

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. This form makes it easy to identify the slope and where the line crosses the y-axis. For y = 2x - 4, the y-intercept is -4, meaning the line crosses the y-axis at (0, -4).

Graphing Linear Equations

Graphing a linear equation involves plotting the y-intercept and using the slope to find additional points. Starting at (0, b), move vertically and horizontally according to the slope to plot points, then draw a straight line through them. This visualizes the relationship between x and y defined by the equation.