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, with equations y = a|x|, y = b|x|, y = c|x|, y = d|x|
(a) for each coefficient, choose whether it is positive or negative.
a: positive (dropdown), b: positive (dropdown), c: negative (dropdown), d: negative (dropdown)
(b) choose the coefficient with the greatest value.
○ a ○ b ○ c ○ d
(c) choose the coefficient closest to 0.
○ a ○ b ○ c ○ d

Explanation:

Part (b)

Step1: Recall absolute value function properties

The general form of the absolute value function is \( y = k|x| \). If \( k>0 \), the graph opens upward; if \( k < 0 \), it opens downward. The magnitude of \( |k| \) determines the "steepness" of the graph: a larger \( |k| \) means a steeper graph.

Step2: Analyze the graphs

• For \( y = A|x| \) and \( y = B|x| \), both open upward (so \( A>0, B>0 \)). The graph of \( y = B|x| \) is steeper than \( y = A|x| \), so \( |B| > |A| \), and since both are positive, \( B > A \).
• For \( y = C|x| \) and \( y = D|x| \), both open downward (so \( C<0, D<0 \)). The graph of \( y = C|x| \) is steeper (more "negative" in terms of steepness, but we compare magnitudes). The magnitude \( |C| \) is larger than \( |D| \), but since they are negative, a larger magnitude of a negative number means a smaller value (e.g., \( -2 < -1 \)). So \( D > C \) (because \( |D| < |C| \), so \( -|D| > -|C| \)).

Step3: Compare all coefficients

Now we have: \( B > A > 0 \), and \( 0 > D > C \). So among \( A, B, C, D \), \( B \) is positive and has a larger magnitude than \( A \), and \( D \) and \( C \) are negative (so smaller than positive numbers). Thus, \( B \) has the greatest value.

Step1: Recall the concept of "closeness to 0"

A number is closer to 0 if its absolute value is smaller. So we need to find the coefficient with the smallest absolute value.

Step2: Analyze the steepness (related to absolute value)

• The graph of \( y = D|x| \) is the least steep among all four graphs. A less steep graph of \( y = k|x| \) means that \( |k| \) is smaller (since steepness is related to \( |k| \)).
• For \( y = A|x| \) and \( y = B|x| \) (upward opening), \( y = D|x| \) (downward opening) is less steep than both \( A \) and \( B \) (since \( B \) is steeper than \( A \), and \( D \)'s graph is flatter than \( A \)'s). Also, compared to \( C \)'s graph (which is steeper than \( D \)'s), \( D \)'s graph is flatter. So \( |D| \) is the smallest among \( |A|, |B|, |C|, |D| \), meaning \( D \) is closest to 0.

Answer:

B

Part (c)