find x intercepts of equation witht the vertex which is (4,-1) and y intercept which is (0,15) . Write answer in (x1,y1),(x2,y2)

Given that we have the vertex \((4, -1)\) and the y-intercept \((0, 15)\), we can find the equation of the parabola and then solve for the x-intercepts.

1. **Equation of the parabola**:
The general form of a parabola with a vertex \((h, k)\) is:

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

In this case, the vertex is \((4, -1)\), so the equation becomes:

\[
y = a(x - 4)^2 - 1
\]

2. **Using the y-intercept to find \(a\)**:
We know that the parabola passes through the point \((0, 15)\), which is the y-intercept. Substituting \(x = 0\) and \(y = 15\) into the equation:

\[
15 = a(0 - 4)^2 - 1
\]

This simplifies to:

\[
15 = a(16) - 1
\]
\[
16a = 16
\]
\[
a = 1
\]

Thus, the equation of the parabola is:

\[
y = (x - 4)^2 - 1
\]

3. **Finding the x-intercepts**:
To find the x-intercepts, we set \(y = 0\) and solve for \(x\):

\[
0 = (x - 4)^2 - 1
\]

Add 1 to both sides:

\[
1 = (x - 4)^2
\]

Take the square root of both sides:

\[
x - 4 = \pm 1
\]

This gives us two solutions:

\[
x - 4 = 1 \quad \text{or} \quad x - 4 = -1
\]

Solving for \(x\):

\[
x = 5 \quad \text{or} \quad x = 3
\]

4. **x-intercepts**:
The x-intercepts are \((5, 0)\) and \((3, 0)\).

Thus, the x-intercepts are:
\[
(5, 0), (3, 0)
\]

Would you like any further details or clarification?

### Related Questions:
1. How do you derive the equation of a parabola given any three points?
2. How do you find the focus and directrix of a parabola?
3. Can a parabola have only one x-intercept?
4. What is the geometric significance of the vertex of a parabola?
5. How does changing the value of \(a\) in the parabola equation affect its graph?

### Tip:
To quickly find the x-intercepts of any parabola, set the equation equal to zero and solve for \(x\) using the quadratic method if necessary.

find x intercepts of equation with the vertex which is (4,-1) and y intercept which is (0,15). Write answer in (x1,y1),(x2,y2)

Learn how to find the x-intercepts of a parabola given its vertex at (4, -1) and y-intercept at (0, 15). This solution covers key quadratic concepts and shows step-by-step how to solve for x-intercepts using the vertex form of a parabola. Suitable for students in Grades 9-10.

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