Finding equations of tangents parallel to and perpendicular to the initial line

Finding Equations of Tangents Parallel to the Initial Line

The equation of a line in two dimensions can be given as y = mx + c, where ‘m’ is the slope of the line and ‘c’ is the y-intercept.
A tangent line to a curve touches the curve at exactly one point. The slope of the tangent line at that point is equal to the derivative of the curve’s equation at that point.
To find the equation of a tangent line to a curve that is parallel to a given line, first find the slope ‘m’ of the given line. Then find the points on the curve where the derivative of the curve’s equation equals ‘m’. Use one of these points and the slope ‘m’ to write the equation of the tangent line.

Two lines are perpendicular if the product of their slopes is -1. Therefore, if a line has slope ‘m’, any line perpendicular to it will have slope ‘-1/m’.
To find the equation of a tangent line to a curve that is perpendicular to a given line, first find the slope ‘m’ of the given line. Then, calculate ‘-1/m’. Afterwards, find the points on the curve where the derivative of the curve’s equation equals ‘-1/m’. Use one of these points and the slope ‘-1/m’ to write the equation of the tangent line.

1. Determine the slope of the initial line.

2. Find the derivative of the function.

3. Set the derivative of the function equal to the slope for a parallel line or ‘-1/m’ for a perpendicular line.

4. Substitute the ‘x’ values into the original function to find the corresponding ‘y’ values - the y-coordinate(s) of the point(s) of tangency.

5. Use the point-slope form of a line (y - y1 = m(x - x1)) to write the equation(s) of the tangent line(s).