Question

1. make sense and persevere the graph shows a jumpers height, y, in feet x seconds after getting onto a trampoline. a. what are the x - and y - intercepts? explain what the x - and y - intercepts represent. b. over what intervals is the graph positive? explain what the positive intervals represent. c. over what intervals is the graph negative? explain what the negative intervals represent. d. choose an interval. find the average rate of change on the interval and interpret its meaning.

Explanation:

Step1: Identify x - intercepts

The x - intercepts are the points where the graph crosses the x - axis. From the graph, the x - intercepts are \(x = 0\), \(x = 0.5\), \(x = 1\), \(x = 1.5\), \(x = 2\), \(x = 2.5\). They represent the times when the jumper's height is 0 feet (i.e., the jumper is on the trampoline surface).

Step2: Identify y - intercept

The y - intercept is the point where the graph crosses the y - axis. Here, \(y = 0\) when \(x = 0\). It represents the initial height of the jumper when they first get onto the trampoline, which is at the surface level (\(y = 0\) feet).

Step3: Find positive intervals

The graph is positive when \(y>0\). The positive intervals are \((0,0.5)\), \((1,1.5)\), \((2,2.5)\) etc. These intervals represent the times when the jumper is above the trampoline surface.

Step4: Find negative intervals

There are no negative intervals for this real - world context since height cannot be negative in this case (\(y\geq0\) always as it represents height above the trampoline).

Step5: Calculate average rate of change

Let's choose the interval \([0,0.5]\). The average rate of change formula is \(\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}\). At \(x = 0,y = 0\) and at \(x = 0.5,y = 1.5\). So \(\frac{\Delta y}{\Delta x}=\frac{1.5 - 0}{0.5-0}=3\). This means that on average, the jumper's height is increasing at a rate of 3 feet per second over the interval \([0,0.5]\) seconds.

Answer:

a. x - intercepts: \(x = 0,0.5,1,1.5,2,2.5\) (jumper on trampoline surface); y - intercept: \(y = 0\) (initial height on trampoline).
b. Positive intervals: \((0,0.5)\), \((1,1.5)\), \((2,2.5)\) etc. (jumper above trampoline surface).
c. No negative intervals (height non - negative).
d. For interval \([0,0.5]\), average rate of change is 3 feet per second (height increasing at 3 ft/s on average in this interval).