Question

graphs of functions
minimum -
maximum -
increasing -
decreasing -
zero -
average rate of change – investigates ______ the outputs are changing
when compared to the inputs
average rate of change
for the function ( y = f(x) ), the average rate that ( f(x) ) changes from ( x = a ) to ( x = b ) is given by:
( \frac{f(b) - f(a)}{b - a} = \frac{\text{how much the } y\text{-values have changed}}{\text{how much the } x\text{-values have changed}} )

Explanation:

Step1: Recall Function Graph Terms

• Minimum: Lowest point (local/global) on the graph. From the graph, the right - most "valley" is the minimum (after the decreasing then increasing part).
• Maximum: Highest point (local/global) on the graph. The left - most "peak" is the maximum (before the decreasing part starts after the first increasing).
• Increasing: Where as \(x\) increases, \(y\) increases (the graph rises from left to right). The first segment (left - most) and the right - most segment are increasing.
• Decreasing: Where as \(x\) increases, \(y\) decreases (the graph falls from left to right). The middle segment (between the two increasing parts) is decreasing.
• Zero: Where the graph crosses the \(x\) - axis ( \(y = 0\) ). The graph crosses the \(x\) - axis at some point (visible near \(x=-1\) and maybe others, but we identify the zero as the \(x\) - intercept).
• Average Rate of Change: The "how fast" (rate) the outputs change relative to inputs, calculated by \(\frac{f(b)-f(a)}{b - a}\), so it investigates "how fast" the outputs are changing.

Step2: Fill in the Blanks

• Minimum: The lowest point on the graph (the right - hand minimum).
• Maximum: The highest point on the graph (the left - hand maximum).
• Increasing: Intervals where as \(x\) increases, \(y\) increases (e.g., \((-\infty, c)\) and \((d, \infty)\) for appropriate \(c,d\) from the graph's shape).
• Decreasing: Interval where as \(x\) increases, \(y\) decreases (e.g., \((c, d)\) from the graph's middle part).
• Zero: The \(x\) - intercept(s) (where \(y = 0\)).
• Average Rate of Change: "how fast" (since it's the rate, so it's about the speed of change of outputs relative to inputs).

Answer:

• Minimum - The lowest point (local minimum) on the graph (the right - most valley).
• Maximum - The highest point (local maximum) on the graph (the left - most peak).
• Increasing - Intervals where the graph rises as \(x\) increases (e.g., left - most and right - most segments).
• Decreasing - Interval where the graph falls as \(x\) increases (middle segment).
• Zero - The \(x\) - intercept(s) (where the graph crosses the \(x\) - axis).
• Average Rate of Change - how fast