Slope-Intercept Form

Def 4a) Slope-Intercept Form

• The Slope-Intercept form refers to writing a line in the following way :

\begin{align} f(x) &= ax + b\\ y &= ax + b \\ y &= mx+b\end{align}

• where $a$ or $m$ refers to the slope of the function and $b$ refers to the y-intercept.

• The 3rd form uses a different coefficient $m$ which is often synonymous with the slope of the function.

• The y-intercept refers to the point that is on the y-axis. It is denoted as: $(0,b)$

The slope would be $m=\frac{1-0}{0-2}=\frac{1}{-2}=-\frac{1}{2}$ The y-intercept is at point $(0,1)$, hence our b = 1 Putting this all together, we can write our line as $y=-\frac{1}{2}x+1$

Below is a picture of an equation of a line:

Slope-Intercept Form to Point-Slope Form

Easily convert any linear equation from a slope-intercept form into a point-slope form, here is the general formula

Example

Suppose I have the following Linear Equation : $y=2x+1$ and I wanted to express it as an equation around the point : $(2,5)$ Then \begin{align*}y &= 2x+1\\ y - 5 &= x + 1 - 5 \\ y - 5 &= 2x -4 \\ y - 5 &= 2(x -2) \end{align*} or you could simply observe that the slope is $2$ and use the Point-Slope Form and arrive to $y-5=2(x-2)$

General Form

Given a 1st degree polynomial of the form : $f(x)=ax+b$, we can express it in terms of a slope and any other point $(x_1,y_1)$, where $f(x_1)=y_1$ by the following way : \begin{align*} f(x) &= ax + b\\ f(x) - y_1 &= ax + b - y_1 \\ f(x) - y_1 &= a(x + \frac{b - y_1}{a}) \\ f(x) - y_1 &= a(x - \frac{y_1 - b}{a}) \\ f(x) - y_1 &= a(x - x_1)\end{align*}

Why does this work?

• For the individuals interested, the reason why this works is because linear equations are injective (Actually they are Bijective). Which means that every single x-value has a unique y-value! You can learn more about the concepts of Injective and Surjective if you want!
• Another Take away is that we can express any y-value in terms of x in the following ways :
• $x=\frac{y-b}{a}$

Slope-Intercept Form to Root Form