# Standard Equation of a Parabola

### Standard Equation of a Parabola:

1. General Form:

• The standard equation of a parabola differs based on its orientation: vertical or horizontal.
• A vertical parabola has the form: $y=a{x}^{2}+bx+c$
• A horizontal parabola has the form: $x=a{y}^{2}+by+c$

2. Components of the Equation:

• $a$:
• Determines the shape and direction of the parabola.
• If $a>0$, the parabola opens upward (for vertical) or rightward (for horizontal).
• If $a<0$, the parabola opens downward (for vertical) or leftward (for horizontal).
• $b$:
• Influences the position of the axis of symmetry and the vertex.
• The axis of symmetry is $x=-\frac{b}{2a}$ for a vertical parabola and $y=-\frac{b}{2a}$ for a horizontal parabola.
• $c$:
• Shifts the parabola vertically (for vertical) or horizontally (for horizontal).

3. Finding the Vertex:

• For a vertical parabola ($y=a{x}^{2}+bx+c$):
• $x=-\frac{b}{2a}$ gives the x-coordinate of the vertex. Substitute into the equation to find the y-coordinate.
• Vertex: $\left(-b\mathrm{/}2a,f\left(-b\mathrm{/}2a\right)\right)$, where $f\left(x\right)$ is the function defining the parabola.
• For a horizontal parabola ($x=a{y}^{2}+by+c$):
• $y=-\frac{b}{2a}$ gives the y-coordinate of the vertex. Substitute into the equation to find the x-coordinate.
• Vertex: $\left(g\left(-b\mathrm{/}2a\right),-b\mathrm{/}2a\right)$, where $g\left(x\right)$ is the function defining the parabola.

4. Axis of Symmetry:

• Vertical Parabola: The axis of symmetry is a vertical line passing through the vertex, given by $x=-\frac{b}{2a}$.
• Horizontal Parabola: The axis of symmetry is a horizontal line passing through the vertex, given by $y=-\frac{b}{2a}$.

5. Orientation and Shape:

• The sign of $a$ determines the orientation and direction of opening for the parabola.
• Positive $a$ opens upward (vertical) or rightward (horizontal).
• Negative $a$ opens downward (vertical) or leftward (horizontal).

### Standard Equation of a Vertical Parabola:

1. Equation Form:

• For a vertical parabola with its axis parallel to the y-axis, the standard form is: $y=a{x}^{2}+bx+c$
• This form represents a parabola that opens either upward or downward.

2. Example:

• Consider the equation: $y=2{x}^{2}-4x+3$
• Identifying the Components:
• Coefficients: $a=2$, $b=-4$, $c=3$
• Vertex Form: 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: $\left(1,1\right)$
• Axis of Symmetry: $x=1$ (line of symmetry for the parabola)
• Direction: Upward (as $a>0$)

3. Plotting the Graph:

• With the vertex at $\left(1,1\right)$, use additional points or the axis of symmetry to draw the parabola.
• Choose values for $x$ to calculate $y$ and plot points to sketch the parabolic curve.

### Standard Equation of a Horizontal Parabola:

1. Equation Form:

• For a horizontal parabola with its axis parallel to the x-axis, the standard form is: $x=a{y}^{2}+by+c$
• This form represents a parabola that opens either rightward or leftward.

2. Example:

• Consider the equation: $x=-3{y}^{2}+6y-4$
• Identifying the Components:
• Coefficients: $a=-3$, $b=6$, $c=-4$
• Vertex Form: 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: $\left(-1,1\right)$
• Axis of Symmetry: $y=1$ (line of symmetry for the parabola)
• Direction: Leftward (as $a<0$)

3. Plotting the Graph:

• With the vertex at $\left(-1,1\right)$, use additional points or the axis of symmetry to draw the parabola.
• Choose values for $y$ to calculate $x$ and plot points to sketch the parabolic curve.