Question

1. use structure sketch a graph given the following key features. domain: (-4, 4) range: (-4, 6 increasing: (-4, 1) decreasing: (1, 4) x-intercepts: (-2, 0), (3, 0) y-intercept: (0, 4) negative: (-4, -2) and (3, 4) positive: (-2, 3)

Explanation:

Step1: Set up coordinate system

Draw x - axis and y - axis. Mark the domain limits \(x=- 4\) (open circle) and \(x = 4\) (open circle) on x - axis, range limits \(y=-4\) (open circle) and \(y = 6\) (closed circle) on y - axis.

Step2: Plot intercepts

Plot \(x\) - intercepts \((-2,0)\) and \((3,0)\), \(y\) - intercept \((0,4)\).

Step3: Determine increasing/decreasing

From \(x=-4\) to \(x = 1\), the function is increasing. So the graph rises from near \(x=-4,y=-4\) (open circle) to \(x = 1\) (peak around \(y\leqslant6\)). From \(x = 1\) to \(x = 4\), it's decreasing, so it falls from \(x = 1\) to near \(x = 4,y=-4\) (open circle).

Step4: Determine positive/negative regions

For \(x\in(-4,-2)\cup(3,4)\), \(y<0\) (below x - axis), for \(x\in(-2,3)\), \(y>0\) (above x - axis). Connect the points smoothly, ensuring the increasing/decreasing, sign, and intercepts are satisfied.

Answer:

The graph is a curve (or piece - wise function graph) with open circles at \((-4,-4)\), \((4,-4)\), closed circle at some point near \(y = 6\) (e.g., \((1,6)\) as a peak), passing through \((-2,0)\), \((3,0)\), \((0,4)\), increasing on \((-4,1)\), decreasing on \((1,4)\), negative on \((-4,-2)\) and \((3,4)\), positive on \((-2,3)\). (A rough sketch can be made following these key features)