Question

1. a) increasing: (2, ∞) decreasing: (-∞, 2) b) increasing: (-∞, 1) decreasing: (1, ∞) c) increasing: (0, ∞) decreasing: (-∞, 0) d) increasing: (-∞, -2), (2, ∞) decreasing: (-2, 2)

Explanation:

Step1: Recall the definition of increasing and decreasing functions

For a function \(y = f(x)\), if for any \(x_1

Step2: Analyze the graph

Looking at the parabola (quadratic function \(y = f(x)\)):

• As \(x\) moves from \(0\) to \(\infty\) (right - hand side of the \(y\) - axis), the \(y\) - values of the function are getting larger. For example, if \(x_1 = 1\) and \(x_2=2\), \(f(1)<f(2)\). So the function is increasing on the interval \((0,\infty)\).
• As \(x\) moves from \(-\infty\) to \(0\) (left - hand side of the \(y\) - axis), the \(y\) - values of the function are getting smaller. For example, if \(x_1=-1\) and \(x_2 = 0\), \(f(-1)>f(0)\). So the function is decreasing on the interval \((-\infty,0)\).

We can also use the concept of the derivative (from calculus, where \(y=f(x)=ax^{2}+bx + c\), \(y^\prime=2ax + b\). For the given parabola (assuming \(a>0\) since it opens upwards), \(y^\prime=2ax\). \(y^\prime>0\) when \(x > 0\) (function is increasing) and \(y^\prime<0\) when \(x<0\) (function is decreasing)).

Answer:

C. Increasing: \((0, \infty)\) Decreasing: \((-\infty, 0)\)