91Ó°ÊÓ

Quadratic functions are polynomial functions of degree two. They can be represented in the standard form: \[ f(x) = ax^2 + bx + c \] where:

• a, b, and c are constants
• x is the variable

The coefficient 'a' determines the direction of the parabola. If \( a > 0 \), the parabola opens upward. If \( a < 0 \), it opens downward.
The graph of a quadratic function is always a parabola. Understanding the shape and direction helps in analyzing the behavior of the function. The largest or smallest point on the graph is known as the vertex.

The vertex of a quadratic function is a crucial point and represents the maximum or minimum value of the function. To find the vertex of a quadratic function in standard form \( f(x) = ax^2 + bx + c \), we use the vertex formula. The vertex \((h, k)\) is given by:

• \( h = -\frac{b}{2a} \)
• \( k = f(h) \), so substitute \( h \) back into the function to find \( k \)

For the given function \( f(x) = -x^2 + 3x - 10 \), we identified \( a = -1 \), \( b = 3 \), and \( c = -10 \). Using the vertex formula: \[ h = -\frac{3}{2(-1)} = \frac{3}{2} \].
Substituting \( h \) back into the function to find \( k \): \[ k = f\left(\frac{3}{2}\right) = -\left(\frac{3}{2}\right)^2 + 3\left(\frac{3}{2}\right) - 10 \] \[ k = -\frac{9}{4} + \frac{9}{2} - 10 = -\frac{31}{4} \] Thus, the vertex is at \( \left(\frac{3}{2}, -\frac{31}{4}\right) \).

A quadratic function will either increase on one interval and decrease on another, divided by the vertex. For the quadratic function \( f(x) = ax^2 + bx + c \):

• If \( a > 0 \), the parabola opens upward, and the function decreases on \( (-\infty, h) \) and increases on \( (h, \infty) \)
• If \( a < 0 \), the parabola opens downward, and the function increases on \( (-\infty, h) \) and decreases on \( (h, \infty) \)

For \( f(x) = -x^2 + 3x - 10 \), since \( a = -1 \) (i.e., \( a < 0 \)), the parabola opens downward. Thus, the function increases to the left of the vertex and decreases to the right of the vertex.
Hence, the function is increasing on the interval: \[ (-\infty, \frac{3}{2}) \].