Question

in exercises 1 - 6, determine when the function is positive, negative, increasing, or decreasing. then describe the end - behavior of the function. (see example 1.)

Explanation:

Step1: Recall function behavior rules

Positive/negative is determined by $y$ - values above/below $x$ - axis. Increasing/decreasing is based on slope direction (rising/falling from left - to - right).

Step2: Analyze graph 1 ($y = 5x$)

The line $y = 5x$ has a positive slope. It passes through the origin. For $x>0$, $y>0$; for $x < 0$, $y<0$. It is increasing for all real $x$. Positive when $x>0$, negative when $x < 0$, increasing for $x\in(-\infty,\infty)$.

Step3: Analyze graph 2 ($y=\frac{1}{2}x$)

The line $y=\frac{1}{2}x$ has a positive slope. It passes through the origin. Positive when $x>0$, negative when $x < 0$, increasing for $x\in(-\infty,\infty)$.

Step4: Analyze graph 3 ($y=-3x - 1$)

The line $y=-3x - 1$ has a negative slope. The $y$ - intercept is - 1. Positive when $x<-\frac{1}{3}$, negative when $x>-\frac{1}{3}$, decreasing for $x\in(-\infty,\infty)$.

Step5: Analyze graph 4 ($y = 4|x|+2$)

For $x\geq0$, $y = 4x+2$ (positive slope), for $x<0$, $y=-4x + 2$ (positive slope). It is always positive ($y\geq2$). Decreasing for $x\in(-\infty,0)$ and increasing for $x\in(0,\infty)$.

Step6: Analyze graph 5 ($y=|x|+2 - 3$ or $y = |x|-1$)

For $x\geq0$, $y=x - 1$, for $x<0$, $y=-x - 1$. Positive when $x>1$ or $x < - 1$, negative when $-1

Step7: Analyze graph 6 ($y=-2|x|+1 + 3$ or $y=-2|x|+4$)

For $x\geq0$, $y=-2x + 4$, for $x<0$, $y = 2x+4$. Positive when $-22$ or $x < - 2$. Decreasing for $x\in(0,\infty)$ and increasing for $x\in(-\infty,0)$.

Answer:

Graph 1: Positive when $x>0$, negative when $x < 0$, increasing for all $x$.
Graph 2: Positive when $x>0$, negative when $x < 0$, increasing for all $x$.
Graph 3: Positive when $x<-\frac{1}{3}$, negative when $x>-\frac{1}{3}$, decreasing for all $x$.
Graph 4: Always positive, decreasing for $x\in(-\infty,0)$, increasing for $x\in(0,\infty)$.
Graph 5: Positive when $x>1$ or $x < - 1$, negative when $-1Graph 6: Positive when $-22$ or $x < - 2$, decreasing for $x\in(0,\infty)$, increasing for $x\in(-\infty,0)$.