Question

a quadratic function $f(x)$ is hidden from view. you must find all intervals where $f(x)$ is increasing. choose the form of the quadratic function $f(x)$ that you would like to see in order to answer the question most efficiently.

form: factored form $f(x) = -2(x - 3)(x + 5)$

Explanation:

Step1: Find the vertex's x - coordinate

For a quadratic function in factored form \(f(x)=a(x - r_1)(x - r_2)\), the x - coordinate of the vertex (the axis of symmetry) is given by the midpoint of the roots. The roots of \(f(x)=-2(x - 3)(x + 5)\) are \(x = 3\) and \(x=-5\). The midpoint formula for two numbers \(x_1\) and \(x_2\) is \(\frac{x_1 + x_2}{2}\). So, the x - coordinate of the vertex is \(\frac{3+(-5)}{2}=\frac{3 - 5}{2}=\frac{-2}{2}=-1\).

Step2: Determine the direction of the parabola

The coefficient \(a\) in the quadratic function \(f(x)=a(x - r_1)(x - r_2)\) determines the direction of the parabola. Here, \(a=-2\), and since \(a<0\), the parabola opens downwards.

Step3: Find the interval of increase

For a parabola that opens downwards, the function is increasing to the left of the vertex (the axis of symmetry) and decreasing to the right of the vertex. Since the x - coordinate of the vertex is \(x = - 1\), the function \(f(x)\) is increasing on the interval \((-\infty,-1)\).

Answer:

The interval where \(f(x)\) is increasing is \((-\infty,-1)\)