Question

Given the graph of a quadratic​ function, write the equation of the function in vertex form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k.

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enter your response here ​(Simplify your​ answer.)

Answer 1

To write the equation of a quadratic function in vertex form \( f(x) = a(x - h)^2 + k \), you'll need to know the following:

1. The vertex of the parabola (h, k).
2. A point on the graph (if the vertex form is not enough).
3. The direction the parabola opens (upward or downward) to determine the sign of 'a'.

Here's how to derive the equation in vertex form step-by-step:

1. **Identify the Vertex**: From the graph, determine the coordinates of the vertex, \( (h, k) \).

2. **Identify a Point**: Find a point \( (x_1, y_1) \) that lies on the parabola, if possible.

3. **Substitute in Vertex Form**: Substitute \( h \) and \( k \) into the vertex form \( f(x) = a(x - h)^2 + k \).

4. **Solve for 'a'**: Use the coordinates of the point \( (x_1, y_1) \) to find the value of 'a' by plugging \( x_1 \) and \( y_1 \) into the equation and solving for 'a':

\( y_1 = a(x_1 - h)^2 + k \)

5. **Complete the Equation**: Now, you can fully write the function in vertex form.

### Example

If the vertex of the function is \( (2, 3) \) and the point \( (3, 5) \) is on the graph:

1. **Vertex**: \( h = 2, k = 3 \)
2. **Point on graph**: \( (3, 5) \)

Now substituting into the vertex form:

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

This simplifies to:

\[
5 = a(1)^2 + 3
\]

\[
5 = a + 3
\]

Subtracting 3 from both sides gives:

\[
2 = a
\]

Now substitute \( a, h, \) and \( k \) back into the vertex form:

\[
f(x) = 2(x - 2)^2 + 3
\]

This is your final equation in vertex form. Please provide specific vertex coordinates and point(s) on the graph if you wish to get a tailored equation.