Question

which graph shows a function that always has a negative average rate of change?

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). If \(b>a\) (i.e., \(b - a>0\)), for the average rate of change to be negative, we need \(f(b)-f(a)<0\), which means \(f(b)<f(a)\). In other words, as \(x\) increases (\(b > a\)), \(y\) decreases.

Step2: Analyze the first graph (a parabola)

For a parabola (a quadratic function \(y=ax^{2}+bx + c\), \(a
eq0\)), it first decreases and then increases (if \(a<0\)) or first increases and then decreases (if \(a > 0\)). So, it does not always have a negative average rate of change.

Step3: Analyze the second graph (a decreasing - type curve)

As \(x\) increases, \(y\) is always decreasing. Let \(x_10\) and \(f(x_2)-f(x_1)<0\), the average rate of change is negative.

Step4: Analyze the third graph (an increasing - type curve)

As \(x\) increases, \(y\) is always increasing. Let \(x_10\) and \(f(x_2)-f(x_1)>0\)).

Answer:

The second graph (the one that is always decreasing as \(x\) increases) shows a function that always has a negative average rate of change.