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: 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 closest to 0.
○ a ○ b ○ c ○ d

(c) choose the coefficient with the greatest value.

Explanation:

Step1: Recall Absolute Value Graph Properties

The general form of an absolute value function is \( y = k|x| \). If \( k>0 \), the graph opens upward (V - shape opening up); if \( k < 0 \), the graph opens downward (V - shape opening down). The magnitude of \( |k| \) determines the "steepness" of the graph: a larger \( |k| \) means a steeper graph, and a smaller \( |k| \) means a flatter graph (closer to 0).

Step2: Analyze Part (b) - Coefficient Closest to 0

- For \( y = A|x| \) and \( y = B|x| \), the graphs open upward (\( A>0, B > 0 \)). The graph of \( y = B|x| \) is flatter than \( y = A|x| \) (since \( A \)'s graph is steeper), so \( |B| < |A| \).
- For \( y = C|x| \) and \( y = D|x| \), the graphs open downward (\( C<0, D < 0 \)). The graph of \( y = D|x| \) is flatter than \( y = C|x| \) (since \( C \)'s graph is steeper), so \( |D| < |C| \).
- Now, compare the flatness of the upward - opening and downward - opening graphs. The graph of \( y = D|x| \) (downward opening) is flatter than \( y = B|x| \) (upward opening)? Wait, no. Wait, looking at the graphs: The first graph (\( y = A|x| \)) is very steep, the second (\( y = B|x| \)) is less steep (more flat) than \( A \), the third (\( y = C|x| \)) is a downward - opening steep graph, and the fourth (\( y = D|x| \)) is a downward - opening flat graph. Wait, actually, the flatter the graph, the closer \( |k| \) is to 0. Among all the graphs, the graph of \( y = D|x| \) is the flattest? Wait, no, wait the fourth graph ( \( y = D|x| \)): Wait, no, let's re - examine. Wait, the standard \( y = |x| \) has a slope of 1 for \( x>0 \) and - 1 for \( x < 0 \). A flatter graph has a slope with smaller magnitude. The graph of \( y = D|x| \) (the fourth graph) is the flattest among all, so \( |D| \) is the smallest, meaning \( D \) is closest to 0? Wait, no, wait: Wait, the upward - opening graphs: \( B \) is flatter than \( A \), so \( |B| < |A| \). The downward - opening graphs: \( D \) is flatter than \( C \), so \( |D| < |C| \). Now, compare \( |B| \) and \( |D| \). The graph of \( y = D|x| \) (downward opening) is flatter than \( y = B|x| \) (upward opening)? Looking at the graphs, the fourth graph ( \( y = D|x| \)) seems to have the smallest slope magnitude. Wait, no, maybe I made a mistake. Wait, the key is: the flatter the graph, the closer the coefficient is to 0. The graph of \( y = D|x| \) is the flattest, so \( D \) is closest to 0? Wait, no, wait the options are A, B, C, D. Wait, let's think again. The graph of \( y = D|x| \): when \( x = 1 \), what's the value of \( y \)? For a flatter graph, \( y \) is closer to 0 when \( x = 1 \). The fourth graph ( \( y = D|x| \)) is the flattest, so \( |D| \) is the smallest, so \( D \) is closest to 0? Wait, no, wait the problem is about the coefficient (the value of \( k \) in \( y=k|x| \)). So for \( y = D|x| \), since the graph is downward opening, \( D<0 \), and its magnitude \( |D| \) is the smallest among all \( |A|,|B|,|C|,|D| \), so \( D \) is closest to 0. Wait, but let's check again. Wait, the upward - opening graphs: \( B \) is less steep than \( A \), so \( |B| < |A| \). The downward - opening graphs: \( D \) is less steep than \( C \), so \( |D| < |C| \). Now, between \( B \) (positive, \( |B| \)) and \( D \) (negative, \( |D| \)): which has a smaller magnitude? The graph of \( y = D|x| \) is flatter than \( y = B|x| \), so \( |D| < |B| \). So \( D \) is closer to 0.

Step3: Analyze Part (c) - Coefficient with Greatest Value

- The value of a number is determined by its sign and magnitude. For positive numbers, a larger magnitude means a larger number. For negative numbers, a larger magnitude means a smaller number (since, e.g., - 5 < - 2).
- \( A>0 \), \( B>0 \), \( C < 0 \), \( D < 0 \). Among positive numbers, \( A \) has a larger magnitude than \( B \) (since \( y = A|x| \) is steeper than \( y = B|x| \)), so \( A > B \). Among negative numbers, \( D \) has a larger value than \( C \) (since \( |D| < |C| \), so \( D > C \) because - 1 > - 5, for example). Now, compare the positive numbers with the negative numbers. Positive numbers are greater than negative numbers. So among \( A \) (positive, large magnitude) and \( B \) (positive, small magnitude), \( A \) is larger. So the coefficient with the greatest value is \( A \)? Wait, no, wait: Wait, \( A \) is positive and has a large magnitude, \( B \) is positive with a small magnitude, \( C \) and \( D \) are negative. So the greatest value (the largest number) is \( A \)? Wait, no, wait the graph of \( y = A|x| \) is very steep, so \( A \) has a large positive value. \( B \) has a smaller positive value. \( C \) and \( D \) are negative. So the greatest value is \( A \)? Wait, but let's check again. Wait, the first graph (\( y = A|x| \)) is the steepest upward - opening graph, so \( A \) is a positive number with a large magnitude. So \( A \) is greater than \( B \) (since \( A \) is steeper, \( |A|>|B| \), and both positive), and \( A \) and \( B \) are greater than \( C \) and \( D \) (since \( C \) and \( D \) are negative). So the coefficient with the greatest value is \( A \).

Answers:

(b)

Explanation:

To determine which coefficient is closest to 0, we use the property of the absolute - value function \( y = k|x| \): the flatter the graph, the smaller the magnitude of \( k \) (and thus the closer \( k \) is to 0).
- For upward - opening graphs (\( y = A|x|, y = B|x| \)): \( y = B|x| \) is flatter than \( y = A|x| \), so \( |B| < |A| \).
- For downward - opening graphs (\( y = C|x|, y = D|x| \)): \( y = D|x| \) is flatter than \( y = C|x| \), so \( |D| < |C| \).
- Comparing the flatness of all graphs, \( y = D|x| \) is the flattest, meaning \( |D| \) is the smallest. So \( D \) is closest to 0.

Answer (b): D

(c)

Explanation:

- For \( y = A|x| \) and \( y = B|x| \) (positive coefficients, since their graphs open upward), \( y = A|x| \) is steeper than \( y = B|x| \), so \( |A|>|B| \) and \( A > B>0 \).
- For \( y = C|x| \) and \( y = D|x| \) (negative coefficients, since their graphs open downward), \( y = D|x| \) is flatter than \( y = C|x| \), so \( |D| < |C| \) and \( D > C \) (because negative numbers with smaller magnitudes are larger).
- Positive numbers are greater than negative numbers. Among positive numbers, \( A \) has a larger magnitude than \( B \), so \( A \) is the largest.

Answer (c): A

To determine which coefficient is closest to 0, we use the property of the absolute - value function \( y = k|x| \): the flatter the graph, the smaller the magnitude of \( k \) (and thus the closer \( k \) is to 0).

• For upward - opening graphs (\( y = A|x|, y = B|x| \)): \( y = B|x| \) is flatter than \( y = A|x| \), so \( |B| < |A| \).
• For downward - opening graphs (\( y = C|x|, y = D|x| \)): \( y = D|x| \) is flatter than \( y = C|x| \), so \( |D| < |C| \).
• Comparing the flatness of all graphs, \( y = D|x| \) is the flattest, meaning \( |D| \) is the smallest. So \( D \) is closest to 0.

• For \( y = A|x| \) and \( y = B|x| \) (positive coefficients, since their graphs open upward), \( y = A|x| \) is steeper than \( y = B|x| \), so \( |A|>|B| \) and \( A > B>0 \).
• For \( y = C|x| \) and \( y = D|x| \) (negative coefficients, since their graphs open downward), \( y = D|x| \) is flatter than \( y = C|x| \), so \( |D| < |C| \) and \( D > C \) (because negative numbers with smaller magnitudes are larger).
• Positive numbers are greater than negative numbers. Among positive numbers, \( A \) has a larger magnitude than \( B \), so \( A \) is the largest.

Answer:

s:
(b)