> [!summary] A secant line is a line that touches at least two points on a curve. > **Key equations:** > Slope of a secant line: > $ \text{slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h} > $ > Equation of a secant line: $f(x) = f (a )+ \frac{f (a+h) - f (a)}{h} (x-a)$ >[!info]+ Read Time **⏱ 3 mins** # Definition A secant line is defined as a line that touches two points (the line itself can touch more than two points, but our definition just cares about two). By definition, the slope of a secant line is the [[Rate of Change|rate of change]] between two points. > [!note]+ Secant Line Diagram ![[sec_1.png|350]] [^1] Example of a secant line touching a curve at two points, A and B By definition, the slope of a secant line is the [[Rate of Change|rate of change]] of two points. However, we often generalize this in terms of a reference point $a$ and a distance from the reference called $h$ > [!warning] Definition assumptions Using the image above, let point 1 be at point A and point 2 be at point B, where: > $ \begin{array}{c} \text{Point 1}: (a, f(a)) \\ \text{Point 2}: (a+h, f(a+h)) \end{array} > $ So now our definition mathematically is: $ \text{slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h} $ > [!bug] Where did $a$ and $h$ come from? In math, we often like to describe secant lines in terms of a reference point. > So let point A (from the image at the start of this note) be the point $(a,f(a))$, then we assume that our distance from point A and point B is a distance $h$ away. This allows us to define point B as $(a+h, f(a+h))$ ## Equation of a Secant Line > [!warning] Assumptions Will use the following notion to simplify our equation: > - Let $y_{2} = f(x), y_{1} = f(a), x_{2} = x, x_{1} = a$ > - So now our [[Point-Slope Form Equation of a Line|point-slope form ]] of an equation of a line can be written as: $f(x)-f(a) =m(x-a)$ > - Slope is $= \frac{f(a+h) - f(a)}{h}$ Now we can write our equation of a secant line as: $ \begin{array}{c} f(x) - f(a) = \frac{f(a+h) - f(a)}{h} (x-a) \\ \\ \text{We often write it in this notion} \\ f(x) = f(a )+ \frac{f(a+h) - f(a)}{h} (x-a) \end{array} $ --- <!-- Light Mode Newsletter Embed --> <div class="mm-form-light"> <iframe src="https://updates.cyberleadhub.com/widget/form/Y0kpQVpjJQuxEfX59m17" id="inline-Y0kpQVpjJQuxEfX59m17" title="Join Math & Matter Newsletter (Light)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div> <!-- Dark Mode Newsletter Embed --> <div class="mm-form-dark"> <iframe src="https://updates.cyberleadhub.com/widget/form/lbeDLm24VjuaFxhjccA1" id="inline-lbeDLm24VjuaFxhjccA1" title="Join Math & Matter Newsletter (Dark)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div> <!-- Provider script (only once) --> <script src="https://updates.cyberleadhub.com/js/form_embed.js"></script> [^1]: Adapted from https://www.integral-domain.org/lwilliams/Resources/TikzImg/secant.tex