~The linear function: there are many equation forms for a straight line. The one which is most popular & convenient (in my opinion), in higher mathematics, is point-slope form: y - y1 = m (x - x1), where (x1,y1) is any point on the line and m is the slope of the line.
~Other forms, such as y = mx + b (slope-intercept), Ax + By + C = 0 (general form), two-pt form, determinant form, & normal form are also used (on occasions, when required by a specific problem), but not as much.
~Notice that all linear equations do not have exponents on x & y greater than one. However, be careful, an equation could contain higher powers of x & y and could define more than one line.
For example: (x-y+2)(3x+5y-7)=0 defines two intersecting lines. If the left side were multiplied out, terms with exponents of 2 would appear and the 2nd degree term xy would also be present.
TANGENT LINES
~To write an equation of a tangent line, one needs to know its slope & a fixed point on the line (using point-slope form).
~The slope is found by calculating the derivative of y with respect to x and substituting the coordinates (usually just x, sometimes just y, & sometimes both x & y) into the derivative.
~A major mistake made by beginning calculus students is to place the derivative (as a variable expression) into the equation form for the line. This is very wrong & destroys the linear form. So, make sure you realize that the slope is a constant (derivative must be evaluated at that point).
Ex: Find an equation of the tangent line to the curve f(x) = x2 at the point (-2,4). First, find f'(x)=2x. Then, substitute x=-2 into the derivative to find the slope of the line: f'(-2)= -4. Now, use point-slope form: y - 4 = -4 [x - (-2)] or y-4=-4(x+2). Beginning students seem to have a difficult time leaving the answer this way. Many want to multiply across the parenthesis & solve for y. (completely unnecessary, in most cases)
NORMAL LINES
~Once you have mastered writing the equation of a tangent line, you are ready for the normal line.
~Simply stated, it is a line perpendicular to the tangent line at the point of tangency. That means it contains the same point as the tangent line. Therefore, the only difference is its slope.
~From basic math, the slopes of perpendicular lines are related by negative reciprocals (some teachers prefer to say , "the product of their slopes is -1"). Either way is fine. So, if the tangent slope is 2, then the normal slope is -1/2. Be careful when the tangent slope is 1 or -1, the normal would seem to be its negative (but actually is the negative reciprocal).
~So, once the derivative is calculated at that point, you can use point-slope form to state both equations quickly (they will look the same except for their slopes).
~Normal lines have wide applications in math & science and usually are used in conjunction with vectors (calculus II & specially calculus III). Normal components of forces play an important part in the motion of particles and normals to surfaces are also studied in calculus III.
~Two curves are said to be Orthogonal if they intersect at right angles. Which means, their tangent lines at their intersection points are perpendicular. To show this, one needs to calculate the derivatives of each curve and substitute the coordinates of each point of intersection into these derivatives. If the resultant numbers are negative reciprocals, then the curves are said to be orthogonal. "Orthogonal Trajectories" are related to force fields & energy and are studied in differential equations.