Question

graph the line that has a slope of \\(\frac{1}{4}\\) and includes the point (4, 8). click to select points on the graph.

Explanation:

Step1: Recall the point - slope form

The point - slope form of a linear equation is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. Here, $m = \frac{1}{4}$ and $(x_1,y_1)=(4,8)$. Substitute these values into the formula:
$y - 8=\frac{1}{4}(x - 4)$

Step2: Simplify the equation

Expand the right - hand side: $y - 8=\frac{1}{4}x-1$
Then, add 8 to both sides of the equation: $y=\frac{1}{4}x - 1+8=\frac{1}{4}x + 7$

Step3: Find two points on the line

We know that the point $(4,8)$ is on the line. To find another point, we can use the slope. The slope $m=\frac{1}{4}=\frac{\text{rise}}{\text{run}}$, which means for a run (change in $x$) of 4 units, the rise (change in $y$) is 1 unit.
Starting from the point $(4,8)$, if we increase $x$ by 4 (so $x = 4 + 4=8$), then $y$ will increase by 1 (so $y=8 + 1 = 9$). So the point $(8,9)$ is on the line.
If we decrease $x$ by 4 from $(4,8)$ (so $x=4-4 = 0$), then $y$ will decrease by 1 (so $y = 8-1=7$). So the point $(0,7)$ is on the line.

Step4: Plot the points and draw the line

Plot the points $(4,8)$, $(8,9)$ and $(0,7)$ on the coordinate plane and draw a straight line passing through them.

Answer:

To graph the line, plot the points \((4,8)\), \((0,7)\) (or \((8,9)\)) and draw a straight line through them. The equation of the line is \(y=\frac{1}{4}x + 7\). The key points on the line are \((4,8)\), \((0,7)\) and \((8,9)\) (among others) which can be used to draw the line on the given coordinate grid.