# Tangent and Secant Lines > [!warning]- English Definitions of "Secant" and "Tangent" > **Tangent:** A straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point. > > **Secant:** In terms of geometry, a secant is a straight line that cuts a curve in two or more parts. > > ![[Pasted image 20220521122800.png]] The [[Rates of Change]] of functions is one of the most crucial concepts within calculus. - A linear function, such as $f(x)=-2x-3$, would have a constant slope - A nonlinear function, such as $f(x)=x^2$, will have *different* rate of change at certain points of the graph. > [!summary]+ Definition > The **secant line** of the function $f(x)$ is a line passing through two points on the function, $(a, f(a))$ and $(x, f(x))$. It's slope is given by: > $m_\text{sec}=\frac{f(x)-f(a)}{x-a}$ > ![[Pasted image 20220521022600.png]] The accuracy of estimating the function's rate of change using a secant line depends on how close $x$ is to $a$. The image below shows two different secant lines ![[Pasted image 20220521022303.png|400]] > It's important to note that **there is no singular secant/tangent line for a function**. Both lines depend entirely on the placement of the $a$ and $x$ points. > > For an example of a secant line changing with different points, take a look at [this interactive demo](https://mathinsight.org/applet/secant_line_slope). As the value of $x$ gets closer and closer to the value of $a$, the secant line approaches something known as the **tangent line** of the original function $f(x)$. However, if $x$ was to equal $a$ the denominator would be $0$, making the equation **undefined**. To find the tangent line we can use a concept known as [[Limits]]. With limits we look at a function's behavior before and after a specific point, in this case $f(a)$. > Remember, when finding the slope of a tangent line $m_\text{tan}=\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$ > [!summary]+ The Difference Quotient > The Difference Quotient is an alternative form of the equation for a tangent/secant line. > > While it's more complex, it allows for you to calculate the value of $m_\text{tan}$ without needing to find the approaching values through the table method. > > If we let $h=x-a$, then $x=a+h$ and $h \to 0$ as $x \to a$. > We can rewrite the limit as: > > $m_\text{tan}=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$ > - [ ] TODO: mention why h is used to represent the difference between points x and a and how it is useful