Question

How do I use the graph of a linear function to find its equation?

Answer 1

Hint: Given the graph of a line, you can determine the equation in two ways, using slope-intercept form, as $$y=mx+b$$ or point slope form as $$y-y_1=m\left(x-x_1\right)$$ . Hence, the slope and one point on the line is all that is needed to write the equation of a line.
Formula used:
 $$m={\displaystyle \frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}}={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$$
 $$y=mx+b$$
where,
m is the slope of a line.
b is the y-intercept

Complete step-by-step answer:
You want to find the equation $$y=mx+b$$ , where m is called the slope, and b the y-value at the y-intercept.
To calculate the slope m: Let's consider the two points selected as $$\left(x_1,y_1\right)$$ and $$\left(x_2,y_2\right)$$ . The slope is how fast y changes relative to a change in x i.e.,
 $$m={\displaystyle \frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}}={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$$ ………………… 1
To calculate the intercept b: If one of the points you selected is on the y-axis, then this y-value is equal to b. Otherwise you fill in the equation $$y=mx+b$$ with the values for one of the points as:
 $$y_1=m{x}_1+b$$
 $$\to b=y_1-m{x}_1$$
Now, substitute the equation of slope i.e., m from equation 1 as:
 $$=y_1-{\displaystyle \frac{y_2-y_1}{x_2-x_1}}{x}_1$$
Now, let us explain using an example:
The points are: $$\left(-4,-2\right)$$ and $$\left(4,0\right)$$ .
We need to find slope m as:
 $$m={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$$
We have the points: $$\left(x_1,y_1\right)=\left(-4,-2\right)$$ and $$\left(x_2,y_2\right)=\left(4,0\right)$$ , hence substituting in the formula we get the slope as:
 $$\Rightarrow m={\displaystyle \frac{0-\left(-2\right)}{4-\left(-4\right)}}$$
 $$\Rightarrow m={\displaystyle \frac{2}{8}}$$
 $$\Rightarrow m={\displaystyle \frac{1}{4}}$$
To calculate the intercept b: Substitute the value of x and y as,
 $$y=mx+b$$
 $$\Rightarrow 0={\displaystyle \frac{1}{4}}\cdot 4+b$$
 $$\Rightarrow 0=1+b$$
 $$\Rightarrow b=-1$$
Substituting the slope and y-intercept into the slope-intercept form of a line the complete equation as:
 $$\to y={\displaystyle \frac{1}{4}}x-1$$
So, the correct answer is “ $$\to y={\displaystyle \frac{1}{4}}x-1$$ ”.

Note: The key point to find its equation using graph is that, we need to Identify the y-intercept of an equation, next we need to choose two points to determine the slope, then substitute the y-intercept and slope into the slope-intercept form of a line. If you have two points on the graph, you can easily calculate the equation i.e., by selecting the two points on the graph and forming the type of equation.