Question

Give the y-intercept and slope for the line. $$y=2 x+1$$

Short answer

Answer: The slope (m) is 2, and the y-intercept (b) is 1 (or the point (0, 1)).

Step-by-step solution

Identify the slope (m)

In the given equation $$y=2x+1$$, the slope is the coefficient of the x-term, which is $$2$$. So, the slope is $$m=2$$.

Identify the y-intercept (b)

In the given equation $$y=2x+1$$, the y-intercept is the constant term, which is $$1$$. So, the y-intercept is $$b=1$$, which means the line intersects the y-axis at the point $$(0,1)$$.

Present the slope and y-intercept

The slope of the line is $$m=2$$, and the y-intercept is $$b=1$$ (or the point $$(0,1)$$).

Key concepts

Slope

The slope of a line is a measure of its steepness or the angle it makes with the horizontal axis. You can visualize it as how much the line rises or falls as you move from left to right along the line. In mathematics, the slope is most commonly represented by the letter 'm'.

For the equation of a line in slope-intercept form, which looks like \( y = mx + b \), the 'm' represents the slope. It shows the ratio of the rise (the change in 'y') to the run (the change in 'x') between any two points on the line. To give you a simple example, a slope of 2 means that for every unit you move to the right (the run), the line rises by 2 units (the rise).

So when you come across an equation like \( y = 2x + 1 \), you can quickly identify the slope by looking at the number in front of the 'x', which is 2 in this case. That means if you move one step to the right on the graph, you will move two steps up. If the slope were negative, you'd move down instead.

Y-intercept

The y-intercept is the point at which the line crosses the y-axis in a coordinate plane. It's where the line 'intercepts' the axis and this point has an 'x' value of 0 because it's directly above or below the origin, depending on whether the y-intercept is positive or negative, respectively. The y-intercept is represented by the letter 'b' in the slope-intercept form of a line, \( y = mx + b \).

In our example, \( y = 2x + 1 \), the constant term '1' is the y-intercept, which makes the point (0, 1) the place where the line touches the y-axis. This point is crucial when graphing the line because it serves as a starting reference point. Once you plot the y-intercept on your graph, you can use the slope to find other points and draw the line.

Equation of a Line

An equation of a line provides a relationship between the x and y coordinates of every point on the line. One common form of such an equation is the slope-intercept form, which is written as \( y = mx + b \), where 'm' is the slope, and 'b' is the y-intercept. This form is incredibly user-friendly because it allows for the direct plotting of the y-intercept and the slope on a graph.

To graph a line using the equation \( y = 2x + 1 \), for example, you start by plotting the y-intercept at (0, 1). After that, you apply the slope '2', which means from your starting point, you move up 2 units and to the right 1 unit to plot another point on the line. By connecting these points, you'll form the line that represents the equation. It's worth noting that, due to the structure of the equation, this method can be used for any line as long as it's in slope-intercept form.