Determine the equation of a quadratic function that has a y intercept of 22, axis of symmetry
of x = 3, and passing through the point (2, 6). Write the vertical stretch in the first box and
the minimum in the second box.

We are tasked with finding the equation of a quadratic function that satisfies the given conditions: a y-intercept of 22, axis of symmetry at \(x = 3\), and passing through the point \((2, 6)\).

### Step 1: General form of the quadratic function

We will use the vertex form of a quadratic equation, which is:

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

Where:
- \(a\) is the vertical stretch (scaling factor),
- \(h\) is the x-coordinate of the vertex (axis of symmetry),
- \(k\) is the y-coordinate of the vertex (minimum value if the parabola opens upwards).

Since the axis of symmetry is \(x = 3\), the vertex is at \((3, k)\).

Thus, the equation becomes:

\[
y = a(x - 3)^2 + k
\]

### Step 2: Use the y-intercept to find \(k\)

The y-intercept occurs when \(x = 0\). At the y-intercept, \(y = 22\), so we substitute this into the equation:

\[
22 = a(0 - 3)^2 + k
\]

Simplifying:

\[
22 = 9a + k
\]

This is our first equation.

### Step 3: Use the point (2, 6) to find another equation

Since the quadratic passes through the point \((2, 6)\), we can substitute \(x = 2\) and \(y = 6\) into the equation:

\[
6 = a(2 - 3)^2 + k
\]

Simplifying:

\[
6 = a(1)^2 + k
\]

\[
6 = a + k
\]

This is our second equation.

### Step 4: Solve the system of equations

We now have the following system of equations:

1. \( 22 = 9a + k \)
2. \( 6 = a + k \)

We can solve this system by substitution or elimination. Let's subtract equation (2) from equation (1):

\[
(22 - 6) = (9a + k) - (a + k)
\]

Simplifying:

\[
16 = 8a
\]

\[
a = 2
\]

### Step 5: Solve for \(k\)

Now that we know \(a = 2\),

Determine the equation of a quadratic function that has a y intercept of 22, axis of symmetry of x = 3, and passing through the point (2, 6). Write the vertical stretch in the first box and the minimum in the second box.

Learn how to find the equation of a quadratic function with a y-intercept of 22, axis of symmetry at x = 3, and passing through the point (2, 6). This problem demonstrates using the vertex form and solving systems of equations to determine the vertical stretch and minimum value of the function. Suitable for Grades 9-11.

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