Question

determine the vertex form equation for the following parabolas.

1. a quadratic function has a vertex at (3,2) and passes through the point (5,-2).

Explanation:

Identify the vertex form equation

We need to find the vertex form of a quadratic function.
The standard vertex form of a parabola is:

$$y = a(x - h)^2 + k$$

where \((h, k)\) represents the coordinates of the vertex.
We are given the vertex \((h, k) = (3, 2)\).
Substituting these values into our equation yields:

$$y = a(x - 3)^2 + 2$$

Substitute the given point to find the leading coefficient

To find the value of the leading coefficient \(a\), we use the given point \((5, -2)\) that lies on the parabola.
We substitute \(x = 5\) and \(y = -2\) into the equation:

$$-2 = a(5 - 3)^2 + 2$$

Solve for the coefficient a

Now we simplify the equation to solve for \(a\):

$$-2 = a(2)^2 + 2$$

$$-2 = 4a + 2$$

Subtract 2 from both sides:

$$-4 = 4a$$

Divide by 4:

$$a = -1$$

Write the final quadratic equation

Now we substitute the value of \(a = -1\) back into our vertex form equation:

$$y = -(x - 3)^2 + 2$$

This represents the final equation of the parabola.

Answer:

$$y = -(x - 3)^2 + 2$$