Questions: Graph the quadratic function f(x)=x^2-10x+24. Give the (a) vertex, (b) axis, (c) domain, and (d) range. Then determine (e) the largest open interval of the domain over which the function is increasing and (f) the largest open interval over which the function is decreasing.

Solution Steps

The vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \) is given by the formula: \[ x = -\frac{b}{2a} \] For the function \( f(x) = x^2 - 10x + 24 \), we have \( a = 1 \), \( b = -10 \), and \( c = 24 \).

Calculate the x-coordinate of the vertex: \[ x = -\frac{-10}{2 \times 1} = \frac{10}{2} = 5 \]

Substitute \( x = 5 \) back into the function to find the y-coordinate: \[ f(5) = 5^2 - 10 \times 5 + 24 = 25 - 50 + 24 = -1 \]

Thus, the vertex is \( (5, -1) \).

The axis of symmetry for a quadratic function is the vertical line that passes through the vertex. Therefore, the axis of symmetry is: \[ x = 5 \]

The domain of any quadratic function is all real numbers: \[ \text{Domain: } (-\infty, \infty) \]

Since the parabola opens upwards (as the coefficient of \( x^2 \) is positive), the range is: \[ \text{Range: } [-1, \infty) \]

Final Answer