Enter parametric equations x(t)=at+b, y(t)=ct+d and parameter t to find (x, y)
x(t)a =, b =
y(t)c =, d =
Parameter t =
Result
When t =
x(t) = at + b
x =
y(t) = ct + d
y =
Corresponding Point
Step-by-Step Derivation
Parametric Equation Formula
x(t) = at + b y(t) = ct + d Given t: x = at + b, y = ct + d
Parametric equations describe the relationship between x and y using a parameter t. As t takes different values, (x,y) traces a path in the plane. For linear parametric equations x=at+b, y=ct+d, the path is a straight line.
⚠Note: If a=c=0, then x=b and y=d are constants, and the path is a single fixed point in the plane, independent of t.
What Is a Parametric Equation?
A parametric equation uses an auxiliary variable (parameter) t to simultaneously express two variables x and y. It is an important tool for describing plane curves and motion trajectories.
Line Equation
x=at+b, y=ct+d describes a line with slope c/a (when a≠0). When a=0, it represents a vertical line.
Slope Calculation
dy/dx = (dy/dt)/(dx/dt) = c/a. As long as a≠0, the slope equals c/a and is independent of t.
Convert to Standard Form
Solve for t from x(t) and substitute into y(t) to eliminate the parameter, giving a direct relationship between y and x.
Path Description
As t varies continuously over an interval, (x,y) traces a curve in the plane — an ideal tool for describing motion paths.
💡 Teaching Example: x(t)=2t+1, y(t)=3t-2. When t=2: x=2×2+1=5, y=3×2-2=4, point (5, 4). Slope k=3/2. Standard form: y=(3/2)(x-1)-2.
A parametric equation uses an auxiliary variable (parameter) t to represent both x and y simultaneously. For x(t)=at+b, y(t)=ct+d, as t takes different values, (x,y) generates different points that together form a straight line.
How do you convert parametric equations to standard form?▼
Solve for t from x(t): t = (x-b)/a, then substitute into y(t): y = c·(x-b)/a + d. Simplify to y = (c/a)x + (d - cb/a), which is the standard linear form y in terms of x.
What are the advantages of parametric equations?▼
Parametric equations control x and y with a single variable t, making them ideal for describing motion trajectories (e.g., projectile motion using time t to control both x and y) and circular motion (e.g., x=cos t, y=sin t) — cases that are difficult to express with standard equations.
How do you find the slope from a parametric equation?▼
Using the chain rule for parametric equations, dy/dx = (dy/dt)/(dx/dt) = c/a. For linear parametric equations, the slope equals c/a and is independent of the value of t.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.