Question

for the function shown in the figure below, at what labeled points is the slope of the graph positive? negative? at which labeled point does the graph have the greatest (i.e., most positive) slope? the least slope (i.e., negative and with the largest magnitude)?

positive slope at a, c and d. negative slope at f. greatest slope at d. least slope at b.
positive slope at a and d. negative slope at c and f. greatest slope at a. least slope at f.
positive slope at a and d. negative slope at b, e and f. greatest slope at d. least slope at f.
positive slope at c and d. negative slope at b and f. greatest slope at c. least slope at d.
positive slope at d. negative slope at b and f. greatest slope at d. least slope at f.

Explanation:

Step1: Recall slope - positive/negative concept

The slope of a graph at a point is positive if the function is increasing at that point (going up as we move from left - to - right), and negative if the function is decreasing (going down as we move from left - to - right).

Step2: Analyze positive slope

At point $D$, the function is increasing. Also, at point $C$, the function is at the bottom of a curve and starting to increase. So, points $C$ and $D$ have positive slopes.

Step3: Analyze negative slope

At point $B$, the function is decreasing. At point $F$, the function is decreasing. So, points $B$ and $F$ have negative slopes.

Step4: Analyze greatest and least slopes

The slope of a graph at a point is related to the steepness of the tangent line at that point. The steeper the positive - sloped tangent line, the greater the slope. Point $D$ has a steeper positive - sloped tangent line compared to $C$. The steeper the negative - sloped tangent line, the least (most negative) the slope. Point $F$ has a steeper negative - sloped tangent line compared to $B$.

Answer:

Positive slope at $C$ and $D$. Negative slope at $B$ and $F$. Greatest slope at $D$. Least slope at $F$.