Curve sketching, linear graphs and slopes

The Inclination of a line and the slope of a line

What is Inclination?
The angle at which the line makes with the positive direction of the x-axis, measured in the anti-clockwise direction, is known as Inclination.
The Inclination is denoted by θ. Shown in Figure (1). It mentioned that (1) The Inclination of a parallel line to the x-axis or the x-axis itself is 0◦ and (2) The Inclination of a line parallel to the y-axis or itself y-axis is 90◦.

What is a Slope?
The Inclination of a line θ =90◦, then tanθ is called a slope. And it is denoted by m:m =tan θ.
Linear graphs are explained through the appropriate equations of a straight line. There are so many equations of a straight line, all of their first degree in x and y.

Equation of a straight line

A common equation of a straight line is,

\[\displaystyle ax+bx+c=0---(1)\]

Here a and b ≠ 0 and a,b,c are real numbers.

Slope Intercept form

\[\displaystyle y=mx+b----(2)\]

m = slope of the line, b= intercept on the y -axis.

Slope point form

The equation of a line with slope m and passing through the point (x1, y1) is,

\[\displaystyle y-{{y}_{1}}=m\left( {x-{{x}_{1}}} \right)---(3)\]

Two-point form

Equation of a line passing through two points (x1, y1) and (x2, y2)

\[\displaystyle y-{{y}_{1}}=\frac{{{{y}_{2}}-{{y}_{1}}}}{{{{x}_{2}}-{{x}_{1}}}}\left( {x-{{x}_{1}}} \right);{{x}_{2}}\ne {{x}_{1}}---(4)\]

Intercept form

\[\displaystyle \frac{x}{a}+\frac{y}{b}=1---(5)\]

X is for the intercept of line a, and y is for the intercept is b. A slope, m, of a line, is the ratio of the change in y compared with the change in x.

\[\displaystyle m=\frac{{\text{change in y}}}{{\text{change in x}}}=\frac{{dy}}{{dx}}----(6)\]

Here, dy = infinitensimal change in y corresponding to the infinitensimal change in x, dx.

Suppose (x1, y1) and (x2, y2) are two points on the line, and m is the slope of the line then,

\[\displaystyle m=\frac{{{{y}_{2}}-{{y}_{1}}}}{{{{x}_{2}}-{{x}_{1}}}};{{x}_{1}}\ne {{x}_{2}}----(7)\]

Parallel and perpendicular lines

Non-vertical lines are parallel, and only if the slope of two lines are equal is,

\[\displaystyle {{m}_{1}}={{m}_{2}}----(8)\]

When two non-vertical lines are perpendicular, then only the product of their slop is -1, i.e.,

\[\displaystyle {{m}_{1}}.{{m}_{2}}=-1----(9)\]

Curve Sketching

What is displacement?
When shortest distance between an object or particle’s initial and final position is called displacement.
According to the study of a particle, the velocity of time graph and displacement time graph is the simplest graphs.
Velocity is the rate of change of displacement: