Answer:
[tex]y = \displaystyle \frac{3}{2}\, x + 1[/tex].
Step-by-step explanation:
The slope-intercept form of a line should be an equation that looks
[tex]y = m \, x + b[/tex].
In this equation,
- [tex]m[/tex] is the slope of the line, and
- [tex]b[/tex] is the y-coordinate of the y-intercept. That's the point where the line crosses the [tex]y[/tex]-axis.
The y-intercept can be found directly from the graph. The line here crosses the [tex]y[/tex]-axis at the point [tex](0,\, 1)[/tex]. The y-coordinate of that point is [tex]1[/tex] (the second number in the tuple.) As a result, [tex]b = 1[/tex].
Finding the slope [tex]m[/tex] of this line can take a bit of an effort. Given two points on the line [tex](x_1,\, y_1)[/tex] and [tex](x_2,\, y_2)[/tex], the slope [tex]m[/tex] of the line would be [tex]\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}[/tex].
The y-intercept [tex](0,\, 1)[/tex] is indeed a point on the graph of the function. That could well be [tex](x_1,\, y_1)[/tex]. (In other words, [tex]x_1 = 0[/tex] and [tex]y_1 = 1[/tex].)
To get the correct value for [tex]m[/tex], make sure that [tex]x_2[/tex] and [tex]y_2[/tex] are as accurate as possible. Try to take only points that are at the intersection of major gridlines. For example, the point [tex](-2,\, -2)[/tex] is on the intersection of gridline [tex]x = 2[/tex] and [tex]y = 2[/tex]. That ensures that the coordinates are quite precise. Since [tex](-2,\, -2)[/tex] is also on the graph of the function, it could serve as [tex](x_2,\, y_2)[/tex]. That means that [tex]x_2 = -2[/tex] and [tex]y_2 = -2[/tex].
Calculate [tex]m[/tex]:
[tex]\begin{aligned} m &= \frac{y_2 - y_1}{x_2 - x_1} \\ &= \frac{0 - (-2)}{1 - (-2)} \\ &= \frac{2}{3}\end{aligned}[/tex].
Hence, the equation for this function (in slope-intercept form) would be:
[tex]y = \displaystyle \frac{3}{2}\, x + 1[/tex].
