# How to find the intersection of an straight line and a function?

I have this following function:

$$y\ =\ \sqrt{\left|x\right|^{2}}-\cos\left(3x\right)$$

which represents this: And I need to find the intersection between an straight line created from two coordinates ($$ab$$ and $$cd$$) and this formula, something like this: I only need the first intersection point coordinates ($$ef$$) (the closest).

On the internet I only find how to intersect circles with other straight lines. How do I do with a function?

I only have $$ab$$ and $$cd$$ coords and need to find $$ef$$.

• In general if you have 2 functions $f(x), g(x)$, the points of intersection between them occur at the zeros of $f(x)-g(x)$. Sep 27, 2019 at 17:40
• @79037662 how can I represent $ab, cd$ into $g(x)$ ? Sep 27, 2019 at 17:43
• it's a straight line so you can calculate the equation of a line from two points: mathsisfun.com/algebra/line-equation-point-slope.html Sep 27, 2019 at 17:46
• You're asking how to find the equation of a line given two points on it, this website can explain much better than I can: mathsisfun.com/algebra/line-equation-2points.html Sep 27, 2019 at 17:47
• You should set equal the equations of the curves (the straight line and the function) which obtains no analytical result in general and should be solved numerically. Sep 27, 2019 at 18:01

You may use point-slope form of line to find the equation of the line. $${{y-y_1}\over x-x_1} = m$$ $${{y-b}\over x-a} = {{b-d}\over {a-c}}$$ Your equation is $$y=|x|-cos(3x)$$
• Substitute $y$ with what? Sep 27, 2019 at 18:38
• Substitute y with $|x|-cos(3x)$ in the line equation.