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:
 \[m = \dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
 \[y = mx + b\]
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.,
 \[m = \dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\] ………………… 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:
 \[ = {y_1} - \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}{x_1}\]
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:
 \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
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:
 \[ \Rightarrow m = \dfrac{{0 - \left( { - 2} \right)}}{{4 - \left( { - 4} \right)}}\]
 \[ \Rightarrow m = \dfrac{2}{8}\]
 \[ \Rightarrow m = \dfrac{1}{4}\]
To calculate the intercept b: Substitute the value of x and y as,
 \[y = mx + b\]
 \[ \Rightarrow 0 = \dfrac{1}{4} \cdot 4 + b\]
 \[ \Rightarrow 0 = 1 + b\]
 \[ \Rightarrow b = - 1\]
Substituting the slope and y-intercept into the slope-intercept form of a line the complete equation as:
 \[ \to y = \dfrac{1}{4}x - 1\]
So, the correct answer is “ \[ \to y = \dfrac{1}{4}x - 1\] ”.

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.