# Parametric Equation of a Parabola

### Parametric Equation of a Parabola:

1. Parametric Equations:

• Parametric equations define curves using parameterized variables $x$ and $y$ as functions of a third parameter, often denoted as $t$.
• For a parabola, parametric equations are expressed as:
• Vertical Parabola: $x=a{t}^{2}+bt+c$ $y=t$
• Horizontal Parabola: $x=t$ $y=a{t}^{2}+bt+c$

2. Key Components:

• Parameter $t$: Serves as the independent variable that traces the curve.
• Coefficients $a$, $b$, and $c$: Determine the shape, position, and orientation of the parabola.

3. Deriving the Parametric Equations:

• Vertical Parabola: $x=a{t}^{2}+bt+c$ and $y=t$
• The parameter $t$ represents the x-coordinate, while $y$ remains linear, producing a vertical parabola.
• Horizontal Parabola: $x=t$ and $y=a{t}^{2}+bt+c$
• The parameter $t$ represents the y-coordinate, while $x$ remains linear, producing a horizontal parabola.

4. Parameter $t$ Interpretation:

• As the parameter $t$ varies, it traces points on the parabolic curve, generating the shape of the parabola.

• Simplifies tracing complex curves by introducing a single parameter that defines multiple coordinates.
• Enables easy manipulation and animation of the curve by altering the parameter values.

6. Graphical Representation:

• Plotting the parametric equations involves varying $t$ within a specified range to trace points on the parabola.
• Plotting multiple values of $t$ produces a series of points that, when connected, form the parabolic curve.

### Example:

Vertical Parabola: Let's consider a vertical parabola defined by the equation $y=2{x}^{2}$. We'll express this parabola in parametric form.

Parametric Equations for Vertical Parabola: For a vertical parabola, the parametric equations are: $x=t$ $y=2{t}^{2}$

Here, the parameter $t$ allows us to trace points on the parabolic curve. As $t$ varies, the corresponding values of $x$ and $y$ generate coordinates that form the parabola.

Horizontal Parabola: Now, let's consider a horizontal parabola defined by the equation $x=3{y}^{2}$. We'll express this parabola in parametric form as well.

Parametric Equations for Horizontal Parabola: For a horizontal parabola, the parametric equations are: $x=3{t}^{2}$ $y=t$

Similar to the vertical parabola, the parameter $t$ allows us to trace points on the curve. Varying $t$ generates different coordinates along the parabolic curve.

Graphical Representation:

• For the vertical parabola, by choosing various values of $t$ within a specific range, we can plot points $\left(t,2{t}^{2}\right)$ to trace the parabolic curve.
• For the horizontal parabola, by choosing values of $t$ and plotting points $\left(3{t}^{2},t\right)$, we can trace the parabolic curve.

Visualizing the Curves:

• Varying the parameter $t$ within a suitable range will generate points on the parabolas.
• By connecting these points, we can visualize and plot the corresponding vertical and horizontal parabolic curves on a graph.

### Vertical Parabola:

Given the parametric equations: $x=2{t}^{2}+3t-1$

$y=t$

Key Components:

• Parameter $t$ represents the independent variable.
• Coefficients $2$, $3$, and $-1$ determine the shape and position of the parabola.

Interpretation:

• As $t$ varies, the values of $x$ and $y$ change, tracing points on the parabolic curve.

Graphical Representation:

• Plotting values of $x$ and $y$ for different $t$ within a specified range will generate points on the parabola.
• As $t$ varies, these points form the parabolic curve.

### Horizontal Parabola:

Given the parametric equations: $x=t$

$y=-{t}^{2}+4t+2$

Key Components:

• Parameter $t$ serves as the independent variable.
• Coefficients $-1$, $4$, and $2$ determine the shape and position of the parabola.

Interpretation:

• $t$ controls the values of $x$ and $y$, generating points on the parabolic curve.

Graphical Representation:

• Varying $t$ within a specified range will produce different $x$ and $y$ values, tracing points on the parabola.
• Connecting these points obtained for various $t$ values will reveal the shape of the parabolic curve.