Introduction to Parabola Equations

Parabolas are fundamental in mathematics, appearing in various applications from optics to engineering. Understanding how to find the equation of a parabola given specific points is crucial. This article will explore how to find the equation of a parabola with a vertex at (-2, 4) and a y-intercept at 2.

Understanding the Vertex Form of a Parabola

The vertex form of a parabola is given by:

[y a(x - h)^2 k]

where (h, k) is the vertex of the parabola. In this case, the vertex is (-2, 4), so we can substitute h and k with their respective values:

[y a(x 2)^2 4]

Using the Y-Intercept to Solve for 'a'

The y-intercept of a parabola is the point where the graph intersects the y-axis. Given the y-intercept is 2, we set x 0 and y 2:

[2 a(0 2)^2 4]

Now, we solve for 'a':

[2 a(2^2) 4]

[2 4a 4]

[2 - 4 4a]

[-2 4a]

[a -frac{1}{2}]

Final Equation of the Parabola

Substituting the value of 'a' back into the vertex form, we get:

[y -frac{1}{2}(x 2)^2 4]

To express this in standard form, we expand the equation:

[y -frac{1}{2}(x^2 4x 4) 4]

[y -frac{1}{2}x^2 - 2x - 2 4]

[y -frac{1}{2}x^2 - 2x 2]

The equation of the parabola is:

[boxed{y -frac{1}{2}x^2 - 2x 2}]

Alternative Approaches and Further Considerations

It's important to note that there can be other forms of parabolas that meet the criteria. For example, the problem can be approached using the standard form [y ax^2 bx c]. In this form, the vertex form can be manipulated to fit the given conditions and solve for the coefficients.

Giventhe vertex is (-2, 4), the equation can be re-written as:

[y a(x 2)^2 4]

Expanding this, gives:

[y a(x^2 4x 4) 4]

[y ax^2 4ax 4a 4]

Given the y-intercept is 2, set x 0 and y 2:

[2 4a 4a 4]

[2 4a 4]

[2 - 4 8a]

[-2 8a]

[a -frac{1}{4}]

Substituting [a -frac{1}{4}] back into the equation:

[y -frac{1}{4}x^2 - x - 2 4]

[y -frac{1}{4}x^2 - x 2]

Thus, another form of the equation that satisfies the given conditions is:

[y -frac{1}{4}x^2 - x 2]

Both the equations [y -frac{1}{2}x^2 - 2x 2] and [y -frac{1}{4}x^2 - x 2] are valid based on the given conditions.

In conclusion, by understanding the vertex form and using the y-intercept, we can determine the equation of a parabola. This method allows for flexibility in solving parabolic equations and can be applied to similar problems with different given parameters.