Question

functions and lines
graphing a line given its slope and y-intercept
graph the line with slope \\(\frac{2}{5}\\) and y-intercept \\(-1\\).

Explanation:

Step1: Recall the slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. Given that the slope $m=\frac{2}{5}$ and the $y$-intercept $b = - 1$, the equation of the line is $y=\frac{2}{5}x-1$.

Step2: Plot the $y$-intercept

The $y$-intercept is $-1$, so we start by plotting the point $(0,-1)$ on the $y$-axis.

Step3: Use the slope to find another point

The slope $m = \frac{2}{5}$ can be thought of as $\frac{\text{rise}}{\text{run}}=\frac{2}{5}$. This means that from the point $(0, - 1)$, we move up 2 units (because the rise is positive) and then move to the right 5 units (because the run is positive). So, from $(0,-1)$, moving up 2 units gives us a $y$-coordinate of $-1 + 2=1$, and moving right 5 units gives us an $x$-coordinate of $0 + 5 = 5$. So the new point is $(5,1)$.

Step4: Draw the line

Now that we have two points, $(0,-1)$ and $(5,1)$, we can draw a straight line passing through these two points. We can also find more points by using the slope in the opposite direction (down 2 units and left 5 units from $(0,-1)$ to get $(-5,-3)$) to make the line more accurate.

Answer:

To graph the line:

1. Plot the point \((0, - 1)\) (the \(y\)-intercept).
2. From \((0, - 1)\), move up 2 units and right 5 units to plot the point \((5,1)\) (using the slope \(\frac{2}{5}=\frac{\text{rise}}{\text{run}}\)).
3. Draw a straight line through the points \((0, - 1)\) and \((5,1)\) (and extend it in both directions).