1. Let's start by understanding what a sigh diagram is. It seems you might mean a "sign diagram," which is used to determine where a function is positive, negative, or zero. 2. The main idea is to find the critical points of a function, which are values where the function is zero or undefined. 3. Then, we test values in the intervals between these critical points to see if the function is positive or negative there. 4. For example, consider the function $f(x) = (x-2)(x+3)$. 5. Step 1: Find zeros by setting $f(x) = 0$: $$ (x-2)(x+3) = 0 $$ which gives $x=2$ or $x=-3$. 6. Step 2: These zeros divide the number line into three intervals: $(-\infty, -3)$, $(-3, 2)$, and $(2, \infty)$. 7. Step 3: Pick test points in each interval, for example, $x=-4$, $x=0$, and $x=3$. 8. Step 4: Evaluate $f(x)$ at these points: - $f(-4) = (-4-2)(-4+3) = (-6)(-1) = 6 > 0$ - $f(0) = (0-2)(0+3) = (-2)(3) = -6 < 0$ - $f(3) = (3-2)(3+3) = (1)(6) = 6 > 0$ 9. Step 5: From this, the sign diagram is: - Positive on $(-\infty, -3)$ - Negative on $(-3, 2)$ - Positive on $(2, \infty)$ 10. This helps us understand where the function is above or below the x-axis. This is the basic process to create and interpret a sign diagram.