P(t) = 0.5t^2 - 9.65t + 100 pt as 200 used in a different form no square root or quadratic formula but using vexter form

To express the quadratic function \( P(t) = 0.5t^2 - 9.65t + 100 \) in vertex form, we can complete the square. The vertex form of a quadratic function is given by:

\[
P(t) = a(t - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola.

1. **Factor out the coefficient of \( t^2 \)**:
   \[
   P(t) = 0.5(t^2 - 19.3t) + 100
   \]

2. **Complete the square**:
   - Take half of the coefficient of \( t \) (which is \(-19.3\)), square it, and add/subtract it inside the parentheses.
   - Half of \(-19.3\) is \(-9.65\), and squaring this gives:
     \[
     (-9.65)^2 = 93.1225
     \]
   - Thus, we rewrite the equation:
   \[
   P(t) = 0.5(t^2 - 19.3t + 93.1225 - 93.1225) + 100
   \]
   - This simplifies to:
   \[
   P(t) = 0.5((t - 9.65)^2 - 93.1225) + 100
   \]

3. **Distribute the 0.5**:
   \[
   P(t) = 0.5(t - 9.65)^2 - 0.5 \times 93.1225 + 100
   \]
   - Calculating \(-0.5 \times 93.1225\):
   \[
   -0.5 \times 93.1225 = -46.56125
   \]
   - So, we have:
   \[
   P(t) = 0.5(t - 9.65)^2 + 53.43875
   \]

The vertex form of the quadratic \( P(t) \) is:

\[
P(t) = 0.5(t - 9.65)^2 + 53.43875
\]

### Summary
- The vertex of the parabola is at \( (9.65, 53.43875) \).
- The function opens upwards (since \( a = 0.5 > 0 \)).

Do you want further details or have any questions? Here are five related questions you might consider:

1. What is the significance of the vertex in a quadratic function?
2. How can you find the axis of symmetry for this quadratic function?
3. What is the maximum or minimum value of this quadratic?
4. How does changing the coefficients affect the graph of the parabola?
5. Can you graph the original and vertex form of the function for comparison?

**Tip**: Completing the square is a valuable technique not just for rewriting quadratics, but also for solving them and analyzing their properties.

P(t) = 0.5t^2 - 9.65t + 100 pt as 200 used in a different form no square root or quadratic formula but using vertex form

Discover how to convert the quadratic function P(t) = 0.5t^2 - 9.65t + 100 into vertex form using the method of completing the square. This step-by-step guide is ideal for high school students learning about quadratic functions.

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