Question

look at the graphs and their equations below. then fill in the information about the coefficients a, b, c, and d.

four graphs of absolute value functions are shown here, each with their equation: y = a|x|, y = b|x|, y = c|x|, y = d|x|

(a) for each coefficient, choose whether it is positive or negative.
a: choose one b: choose one c: choose one d: choose one

(b) choose the coefficient with the greatest value.
○ a ○ b ○ c ○ d

(c) choose the coefficient closest to 0.
○ a ○ b ○ c ○ d

Explanation:

Step1: Sign of coefficients

For the absolute value function $y=k|x|$:

• If $k>0$, the graph opens upward.
• If $k<0$, the graph opens downward.
• Graphs of $y=A|x|$ and $y=B|x|$ open upward: $A>0$, $B>0$
• Graphs of $y=C|x|$ and $y=D|x|$ open downward: $C<0$, $D<0$

Step2: Magnitude of coefficients

The larger $|k|$ is, the steeper the graph.

• Steepest upward graph: $y=B|x|$, so $B>A>0$
• Steepest downward graph: $y=D|x|$, so $|D|>|C|$, meaning $C>D$ (since both negative)
• Order of values: $B>A>C>D$

Step3: Greatest coefficient

From the order $B>A>C>D$, $B$ is the largest.

Step4: Coefficient closest to 0

The least steep graph is $y=C|x|$, so $|C|$ is smallest, making $C$ closest to 0.

Answer:

(a)
A: Positive
B: Positive
C: Negative
D: Negative

(b) B. B

(c) C. C