# Standard Forms of a Parabola

### Standard Forms of a Parabola:

1. Vertical Parabola:

• General Equation: $y=a{x}^{2}+bx+c$
• Standard Form: $y=a\left(x-h{\right)}^{2}+k$ or $\left(x-h{\right)}^{2}=4p\left(y-k\right)$
• Key Components:
• $a$: Determines the direction and width of the parabola's opening.
• $h$: Represents the horizontal shift of the vertex.
• $k$: Indicates the vertical shift of the vertex.
• $p=\frac{1}{4a}$: Distance between the vertex and the focus (if $a>0$) or directrix (if $a<0$).
•

2. Horizontal Parabola:

• General Equation: $x=a{y}^{2}+by+c$
• Standard Form: $x=a\left(y-k{\right)}^{2}+h$ or $\left(y-k{\right)}^{2}=4p\left(x-h\right)$
• Key Components:
• $a$: Determines the direction and width of the parabola's opening.
• $h$: Represents the horizontal shift of the vertex.
• $k$: Indicates the vertical shift of the vertex.
• $p=\frac{1}{4a}$: Distance between the vertex and the focus (if $a>0$) or directrix (if $a<0$).
•

3. Vertex Form:

• Vertical Parabola: $y=a\left(x-h{\right)}^{2}+k$
• Horizontal Parabola: $x=a\left(y-k{\right)}^{2}+h$
• Key Points:
• Represents the parabola with its vertex at $\left(h,k\right)$.
• Describes the parabola's shape and vertex without explicitly showing the focus and directrix.

4. Finding the Focus and Directrix:

• Vertical Parabola: Focus is at $\left(h,k+\frac{1}{4a}\right)$ and directrix is $y=k-\frac{1}{4a}$.
• Horizontal Parabola: Focus is at $\left(h+\frac{1}{4a},k\right)$ and directrix is $x=h-\frac{1}{4a}$.

5. Identifying the Axis of Symmetry:

• For both vertical and horizontal parabolas, the axis of symmetry is a line passing through the vertex and perpendicular to the directrix/focus.

Example: Consider the equation $y=2{x}^{2}-4x+3$

• Identifying Components:
• $a=2$, $b=-4$, $c=3$
• To find the vertex, use $x=-\frac{b}{2a}$:
• $x=-\frac{-4}{2×2}=1$
• Substitute $x=1$ into the equation to find $y$:
• $y=2×{1}^{2}-4×1+3=2-4+3=1$
• Vertex: $\left(1,1\right)$
• $p=\frac{1}{4a}=\frac{1}{8}$
• Focus: $\left(1,1+\frac{1}{8}\right)=\left(1,1.125\right)$
• Directrix: $y=1-\frac{1}{8}=\frac{7}{8}$

Example: Consider the equation $x=-3{y}^{2}+6y-4$

• Identifying Components:
• $a=-3$, $b=6$, $c=-4$
• To find the vertex, use $y=-\frac{b}{2a}$:
• $y=-\frac{6}{2×\left(-3\right)}=1$
• Substitute $y=1$ into the equation to find $x$:
• $x=-3×{1}^{2}+6×1-4=-3+6-4=-1$
• Vertex: $\left(-1,1\right)$
• $p=\frac{1}{4a}=\frac{1}{12}$
• Focus: $\left(-1+\frac{1}{12},1\right)=\left(-\frac{11}{12},1\right)$
• Directrix: $x=-1-\frac{1}{12}=-\frac{13}{12}$

### Example :

Given the equation: $y=2\left(x-3{\right)}^{2}+4$

This equation is in the vertex form of a parabola with the vertex at $\left(3,4\right)$.

Key Components:

• $a=2$ determines the direction and width of the parabola.
• $h=3$ represents the horizontal shift of the vertex.
• $k=4$ indicates the vertical shift of the vertex.

• Axis of Symmetry: $x=h=3$ (vertical line passing through the vertex).
• Focus and Directrix: The focus is at $\left(h,k+\frac{1}{4a}\right)=\left(3,4+\frac{1}{8}\right)=\left(3,4.125\right)$. The directrix is $y=k-\frac{1}{4a}=4-\frac{1}{8}=3.875$.
• The vertex at $\left(3,4\right)$ is the lowest point on the parabola.
• The parabola opens upward because $a>0$.
• The axis of symmetry is a vertical line passing through $x=3$.