Final answer:
To find the equation of a parabola with a vertex and two roots, you can use the vertex form of the equation. The vertex form of a parabola is given by y = a(x-h)² + k, where (h, k) is the vertex of the parabola. Given that the vertex is (2, 5.1215) and the roots are -2 and 7, the equation of the parabola is y = x² - 4x + 9.1215.
Step-by-step explanation:
To find the equation of a parabola with a vertex and two roots, you can use the vertex form of the equation. The vertex form of a parabola is given by y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola.
Given that the vertex is (2, 5.1215) and the roots are -2 and 7, we can start by finding the values of a, h, and k.
- The vertex form gives us the values of h and k directly, so h = 2 and k = 5.1215.
- Next, we can use the fact that the roots are -2 and 7 to find the value of a. Since the roots are -2 and 7, we know that (x+2) and (x-7) are factors of the equation. So, multiplying these two factors should give us the equation of the parabola.
- Multiplying (x+2)(x-7), we get x^2 - 5x - 14.
Substituting the values of a, h, and k into the vertex form, we get the equation of the parabola as y = a(x-2)^2 + 5.1215 = a(x^2 - 4x + 4) + 5.1215 = ax^2 - 4ax + (4a + 5.1215). Since we know that the factor of x^2 in the equation is 1, we can equate this to ax^2 and solve for a.
Equating the coefficients of x^2, we get 1 = a. So the equation of the parabola is y = x^2 - 4x + 4 + 5.1215 = x^2 - 4x + 9.1215.