The vertex of a particular parabola is (4,7). If the parabola also has a 𝑦-intercept of −1, find the sum of the roots for the parabola above.

Given the vertex of the parabola \((4, 7)\) and the \(y\)-intercept \(-1\), we need to determine the sum of the roots of the parabola.

The vertex form of a parabolic equation is:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. Given the vertex \((4, 7)\), the equation becomes:
\[
y = a(x - 4)^2 + 7
\]

Next, we use the \(y\)-intercept to find \(a\). The \(y\)-intercept occurs when \(x = 0\) and \(y = -1\). Substituting \(x = 0\) and \(y = -1\) into the equation:
\[
-1 = a(0 - 4)^2 + 7
\]
Simplify and solve for \(a\):
\[
-1 = 16a + 7
\]
\[
-1 - 7 = 16a
\]
\[
-8 = 16a
\]
\[
a = -\frac{1}{2}
\]

Now, substitute \(a\) back into the equation:
\[
y = -\frac{1}{2}(x - 4)^2 + 7
\]

To find the roots, we need to express the equation in standard form \(ax^2 + bx + c = 0\). Start by expanding the vertex form:
\[
y = -\frac{1}{2}(x^2 - 8x + 16) + 7
\]
\[
y = -\frac{1}{2}x^2 + 4x - 8 + 7
\]
\[
y = -\frac{1}{2}x^2 + 4x - 1
\]

The standard form of the equation is:
\[
-\frac{1}{2}x^2 + 4x - 1 = 0
\]

In a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots is given by \(-\frac{b}{a}\). Here, \(a = -\frac{1}{2}\) and \(b = 4\). Thus:
\[
\text{Sum of the roots} = -\frac{4}{-\frac{1}{2}} = \frac{4}{\frac{1}{2}} = 4 \times 2 = 8
\]

Therefore, the sum of the roots for the parabola is:
\[
\boxed{8}
\]

Would you like more details or have any questions?

Here are 8 related questions:

1. How do you convert from vertex form to standard form of a quadratic equation?
2. What is the significance of the vertex in a parabolic graph?
3. How do you determine the direction of opening for a parabola?
4. How can you find the roots of a quadratic equation given in vertex form?
5. What is the relationship between the coefficients in a quadratic equation and its roots?
6. How do you find the \(x\)-intercepts of a parabola given in standard form?
7. How does the value of \(a\) in the quadratic equation affect the shape of the parabola?
8. What is the axis of symmetry for the parabola given in vertex form?

**Tip:** Knowing the properties of quadratic equations and their forms (vertex and standard) can help in easily finding critical points such as the vertex, axis of symmetry, and roots.

Learn how to find the sum of roots for a parabola given its vertex (4, 7) and y-intercept (-1). This problem involves converting from vertex form to standard form of a quadratic equation, determining the sum of roots using the coefficients, and applying key theorems related to quadratic equations. Suitable for students in Grades 10-12.

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