THE PARABOLA: FINDING VERTEX, FOCUS, DIRECTRIX AND SYMMETRY || CONIC SECTIONS || PRECALCULUS

Summary

This video provides a step-by-step guide on how to determine the vertex, focus, axis of symmetry, and directrix of a parabola given its equation. It also covers sketching the graph and understanding how the orientation of the parabola (upward, downward, left, or right) is determined by its equation.

Highlights

Introduction to Parabola Orientation00:00:00

The video introduces the topic of determining properties of a parabola. It explains that if the equation is x-squared, the parabola opens upward (if positive) or downward (if negative). If the equation is y-squared, the parabola opens to the left (if negative) or to the right (if positive).

Step 1: Rearrange the Equation00:01:01

The first step is to rearrange the given equation. The variable with exponent 2 should be on the left side, and all other terms should be transposed to the right side. In the example, y-squared is the squared term, so it's moved to the left, indicating the parabola opens left or right.

Step 2: Complete the Square00:01:50

The next step involves completing the square for the terms on the left side. This is done by taking the coefficient of the linear term, dividing it by 2, and then squaring the result. This value is then added to both sides of the equation.

Step 3: Factor and Simplify to Standard Form00:02:36

After completing the square, factor the perfect square trinomial on the left side. On the right side, simplify the constant terms. Finally, factor out the coefficient of the variable term on the right side to get the equation into its standard form, (y-k)^2 = 4p(x-h) or (x-h)^2 = 4p(y-k).

Determine Vertex and 'p' Value00:03:55

From the standard equation, identify the vertex (h, k). The video explains how to deduce h and k from the equation. It then shows how to find the value of 'p' by equating 4p with the coefficient of the non-squared term.

Calculate Focus and Directrix00:05:33

Using the vertex (h, k) and the value of 'p', the focus and directrix are calculated. The formulas for focus (h+p, k) and directrix (x = h-p) are applied for a parabola opening to the right. The calculated focus is (7, -4) and the directrix is x = -3.

Graphing and Axis of Symmetry00:07:26

The video then illustrates how to plot the vertex, focus, and directrix on a graph. It explains that the focus must always be inside the parabola, confirming its orientation (opening to the right in this example). Finally, the axis of symmetry is identified as y = k, which is y = -4 in this case.