To determine the standard form of the equation of a parabola given the vertex [tex]\( V(-1, 4) \)[/tex] and focus [tex]\( F(5, 4) \)[/tex], follow these steps:
1. Identify the Orientation of the Parabola:
- The y-coordinates of the vertex and the focus are the same ([tex]\( y = 4 \)[/tex]). This indicates that the parabola is oriented horizontally.
2. Determine the Direction of Opening:
- Since the x-coordinate of the focus ([tex]\( 5 \)[/tex]) is to the right of the x-coordinate of the vertex ([tex]\( -1 \)[/tex]), the parabola opens to the right.
3. Calculate the Distance [tex]\( p \)[/tex]:
- The distance [tex]\( p \)[/tex] between the vertex and the focus is the difference between their x-coordinates.
[tex]\[
p = 5 - (-1) = 6
\][/tex]
4. Determine the Equation of the Directrix:
- For a horizontal parabola, the directrix is a vertical line. The directrix is located at a distance [tex]\( p \)[/tex] to the left of the vertex because the parabola opens to the right.
[tex]\[
x = -1 - p = -1 - 6 = -7
\][/tex]
5. Write the Standard Form of the Equation:
- The standard form of a horizontally oriented parabola with vertex [tex]\((h, k)\)[/tex] and parameter [tex]\( p \)[/tex] is:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
6. Plug in the Values:
- Here, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-1, 4)\)[/tex] and [tex]\( p \)[/tex] is [tex]\( 6 \)[/tex]. So, we substitute these values into the equation:
[tex]\[
(y - 4)^2 = 4 \cdot 6 (x + 1)
\][/tex]
[tex]\[
(y - 4)^2 = 24(x + 1)
\][/tex]
Thus, the standard form of the equation of the parabola is:
[tex]\[
(y - 4)^2 = 24(x + 1)
\][/tex]
