Question

look at the graphs and their equations below. then fill in the information about the coefficients a, b, c, and d.
(a) for each coefficient, choose whether it is positive or negative.
(b) choose the positive or negative coefficient closest to 0.
(c) choose the coefficient with the greatest value.

Explanation:

Step1: Analyze the shape of absolute - value function

The general form of an absolute - value function is $y = k|x|$. When $k>0$, the graph opens upwards, and when $k < 0$, the graph opens downwards.

Step2: Determine the sign of coefficients

For $y = A|x|$, the graph opens upwards, so $A>0$. For $y = B|x|$, the graph opens upwards, so $B>0$. For $y = C|x|$, the graph opens upwards, so $C>0$. For $y = D|x|$, the graph opens downwards, so $D<0$.

Step3: Analyze the steepness of the graph

The larger the absolute value of the coefficient of $|x|$, the steeper the graph. The graph of $y = B|x|$ is less steep than the graph of $y = A|x|$, and the graph of $y = C|x|$ is steeper than the graph of $y = A|x|$. Among positive coefficients $A$, $B$, and $C$, $B$ has the smallest value.

Step4: Find the coefficient closest to 0

Since $D$ is negative and $A$, $B$, $C$ are positive, and $B$ is the smallest among $A$, $B$, $C$, the coefficient closest to 0 is $B$.

Step5: Find the coefficient with the greatest value

Among $A$, $B$, $C$, $D$, since $D$ is negative and $C$ has the steepest positive - opening graph, $C$ has the greatest value.

Answer:

(a)
A: positive
B: positive
C: positive
D: negative
(b) B
(c) C