Pre-Calculus Prep: Conic Sections - Graph a SIDEWAYS Parabola

Certainly! To graph a sideways parabola and determine its focus and directrix, follow these steps:

1. Identify the equation: Let's say we have the equation x = a(y - h)^2 + k, which represents a sideways or horizontal parabola.

2. Determine the vertex: The vertex of the parabola is at the point (h, k). The value of h represents the horizontal shift, and k represents the vertical shift of the parabola.

3. Plot the vertex: On a coordinate plane, plot the vertex at (h, k).

4. Determine the orientation of the parabola: Since we have a sideways parabola, it opens either to the left or to the right. The orientation depends on the sign of a in the equation.

- If a is greater than 0, the parabola opens to the right.
- If a is less than 0, the parabola opens to the left.

5. Determine the focus: The focus of the parabola is a point located at a distance of p units to the right or left of the vertex, where p is equal to 1 divided by 4 times the absolute value of a.

- If the parabola opens to the right (a is greater than 0), the focus will be to the right of the vertex.
- If the parabola opens to the left (a is less than 0), the focus will be to the left of the vertex.

The distance of the focus from the vertex is p units.

6. Determine the directrix: The directrix is a vertical line located at a distance of p units to the right or left of the vertex.

- If the parabola opens to the right (a is greater than 0), the directrix will be a vertical line to the left of the vertex.
- If the parabola opens to the left (a is less than 0), the directrix will be a vertical line to the right of the vertex.

7. Plot the focus and directrix: On the coordinate plane, mark the focus and draw the vertical directrix.

8. Plot additional points: To complete the graph of the parabola, you can select more y-values and calculate the corresponding x-values using the given equation. Alternatively, you can use the symmetry of the parabola to reflect points across the axis of symmetry.

9. Connect the plotted points: Draw a smooth curve passing through the plotted points, which represents the sideways parabola.

By following these steps, you can graph a sideways parabola, determine its focus and directrix, and represent it accurately on a coordinate plane in pre-calculus.

These videos are designed to review and reteach Precalculus and Collegeboard Pre-CALC AP content. My videos cover functions, polynomials, exponential and logarithmic expressions, trigonometry, parametric equations, polar coordinates, vectors, matrices and systems, conic sections, discrete mathematics, sequences and series; and an introduction to calculus.

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa .
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