To derive a parabola's standard equation from its graph, you must first find the parabola's equation in vertex form. Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola and "a" is the positive or negative factor by which the parabola is stretched or compressed. Once you have found the equation of the parabola in vertex form, you can convert this equation to standard form, y = axe^2 + bx + c.

- Skill level:
- Moderate

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## Instructions

- 1
Determine the vertex of the parabola. The lowest or highest point of the parabola is the vertex, and its x and y coordinates can be written in the form (h, k).

- 2
Find the a-value for the graph. First, you want to determine if the a-value is positive or negative. If the parabola opens upward, the a-value is positive. If the parabola opens downward, the a-value is negative. Next, find the numerical value of "a." A standard parabola, a parabola with an a-value of 1, contains the points (-1, 1) and (1, 1). In your graphs, the y-values that correspond to the x-values -1 and 1 will equal the a-value. If the a-value is less than 1, the graph is widened by the inverse of this number. If the a-value is greater than 1, the graph is compressed by this factor. For example, if at the x-values of -1 and 1, the y-values are both 1/4, the parabola is widened by a factor of 4. If the y-values are both 4, the parabola is compressed by a factor of 4.

- 3
Write the equation of the parabola in vertex form. Using the values you found in Steps 1 and 2, plug variable values into the equation y = a(x - h)^2 + k.

## Vertex Form

- 1
Expand the binomial (x - h)^2 by multiplying (x - h) by (x - h). This will give the result x^2 - 2hx + h^2. The parabola's equation is now y = a(x^2 - 2hx + h^2) + k.

- 2
Multiply the polynomial found in Step 1 by the a-value, giving the result axe^2 - 2ahx + ah^2. The parabola's equation is now y = axe^2 - 2ahx + ah^2 + k.

- 3
Substitute the values found in Section 1 into the parabola's equation. The equation should now be in standard form, y = axe^2 + bx + c, where a = a, 2ah = b and (ah^2 + k) = c.