Question

How to Identified maximum and minimum of turning point of parabola

Answer 1

To identify the maximum and minimum points (turning points) of a parabola, you can follow these steps:

1. **Understanding the Parabola**: A parabola can be described by a quadratic function in the form \( y = ax^2 + bx + c \).

2. **Identifying the Shape**:
- If the coefficient \( a \) (the leading coefficient) is positive, the parabola opens upwards, and the turning point is a **minimum**.
- If \( a \) is negative, the parabola opens downwards, and the turning point is a **maximum**.

3. **Finding the Vertex**: The turning point (vertex) of the parabola can be found using the formula:

\[
x = -\frac{b}{2a}
\]

Here, \( b \) is the coefficient of \( x \) in your quadratic equation and \( a \) is the leading coefficient.

4. **Calculating the y-coordinate of the Vertex**: Once you have the x-coordinate of the turning point, substitute it back into the original quadratic equation to find the y-coordinate:

\[
y = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c
\]

This gives you the full coordinates of the vertex \( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \).

5. **Conclusion**:
- If \( a > 0 \): The vertex gives you the minimum point, and the coordinates are \( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \).
- If \( a < 0 \): The vertex gives you the maximum point, and the coordinates remain the same as above.

### Example:

Consider the quadratic function \( y = 2x^2 + 3x + 1 \).

1. **Find the vertex**:
- \( a = 2 \), \( b = 3 \)
- \( x = -\frac{b}{2a} = -\frac{3}{2 \cdot 2} = -\frac{3}{4} \)

2. **Substitute back to find y**:
\[
y = 2\left(-\frac{3}{4}\right)^2 + 3\left(-\frac{3}{4}\right) + 1 = 2\left(\frac{9}{16}\right) - \frac{9}{4} + 1 = \frac{18}{16} - \frac{36}{16} + \frac{16}{16} = \frac{-2}{16} = -\frac{1}{8}
\]

3. **Vertex**: The turning point (minimum point) is \( \left(-\frac{3}{4}, -\frac{1}{8}\right) \).

### Summary:
- Use \( x = -\frac{b}{2a} \) to find the x-coordinate of the turning point.
- Substitute that value into the quadratic equation to find the corresponding y-coordinate.
- Determine whether it is a maximum or a minimum based on the value of \( a \).