Question

(Help) Here's a graph of a linear function. Write the equation that describes that function. (Zoom in if blurry)

Answer 1

Answer:

$y = \displaystyle \frac{3}{2}\, x + 1$.

Step-by-step explanation:

The slope-intercept form of a line should be an equation that looks

$y = m \, x + b$.

In this equation,

• $m$ is the slope of the line, and
• $b$ is the y-coordinate of the y-intercept. That's the point where the line crosses the $y$-axis.

The y-intercept can be found directly from the graph. The line here crosses the $y$-axis at the point $(0,\, 1)$. The y-coordinate of that point is $1$ (the second number in the tuple.) As a result, $b = 1$.

Finding the slope $m$ of this line can take a bit of an effort. Given two points on the line $(x_1,\, y_1)$ and $(x_2,\, y_2)$, the slope $m$ of the line would be $\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$.

The y-intercept $(0,\, 1)$ is indeed a point on the graph of the function. That could well be $(x_1,\, y_1)$. (In other words, $x_1 = 0$ and $y_1 = 1$.)

To get the correct value for $m$, make sure that $x_2$ and $y_2$ are as accurate as possible. Try to take only points that are at the intersection of major gridlines. For example, the point $(-2,\, -2)$ is on the intersection of gridline $x = 2$ and $y = 2$. That ensures that the coordinates are quite precise. Since $(-2,\, -2)$ is also on the graph of the function, it could serve as $(x_2,\, y_2)$. That means that $x_2 = -2$ and $y_2 = -2$.

Calculate $m$:

$\begin{aligned} m &= \frac{y_2 - y_1}{x_2 - x_1} \\ &= \frac{0 - (-2)}{1 - (-2)} \\ &= \frac{2}{3}\end{aligned}$.

Hence, the equation for this function (in slope-intercept form) would be:

$y = \displaystyle \frac{3}{2}\, x + 1$.