Curve_sketching.md

Inclination of a line and the slope of a line

The angle which a line makes with the positive direction of x-axis measured in the anticlockwise direction it is called inclination. (or angle of inclination) Denoted by $\theta$

if $\theta$ $\neq 90\degree$is the inclination of a line, then $\tan\theta$ is called its slope or gradient, usually denoted m : m = tan $\theta$

Equation of Straight Line

1. Genral Equation of Straight Line

$$ ax + by + c = 0 $$

Where a and b $\neq$ 0 and a ,b ,c are real numbers.

2. Slope Intercept form

$$ y = mx + b$$ Where m is the slope of the line and b is the intercept on the y -axis.

For example, if you have an equation in slope-intercept form, such as y = 2x + 3, you can determine that the slope is 2, meaning that for every 1 unit increase in x, y increases by 2 units. The y-intercept is 3, indicating that the line crosses the y-axis at the point (0, 3).

3. Slope - Point Form

The equation of a line with slope m and passing through a point $(x_1, y_1)$ is $$y - y_1 =mx +b $$

4. Two Points form

he equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is

$$\def\a{x_1} \def\b{x_2} \def\c{y_1} \def\d{y_2} y - \c = \frac{\d - \c}{\b - \a}(x - \a); \b \neq \a$$

5. Intercept Form

$$ \frac{x}{a}+\frac{y}{b}=1$$

where x- intercept of the line a and y- intercept is b.

The slope , m of a line is the ratio of the change in y compared with change in x. thus

$$ m = \frac{\text{Chnage in y}}{Change in x}= \frac{\mathrm{d}y}{\mathrm{d}x} $$

If $\def\xa{x_1} \def\xb{x_2} \def\ya{y_1} \def\yb{y_2} (\xa , \ya)\ and (\xb,\yb)\ \text{are two points on a line and m is the slope of the line , then}$

$$ \def\xa{x_1} \def\xb{x_2} m = \frac{y_2 -y_1}{\xa-\xb} ; \xa \neq \xb $$

Parallel and Perpendicular: Two non-vertical lines are parallel if and only if the slopes of the two lines are are equal

$$ m_1 = m_2 $$

Two non-vertical lines are perpendicular if and only if the product of their slopes is -1.