Question

look at the graphs and their equations below. then fill in the information about the coefficients a, b, c, and d.
(the four graphs are shown here, 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: (choose one) b: (choose one) c: (choose one) d: (choose one)
(b) choose the coefficient closest to 0.
○ a ○ b ○ c ○ d
(c) choose the coefficient with the greatest value.
○ a ○ b ○ c ○ d

Explanation:

To solve this problem, we analyze the graphs of the absolute value functions \( y = A|x| \), \( y = B|x| \), \( y = C|x| \), and \( y = D|x| \).

Part (a): Sign of Coefficients

The general form of an absolute value function is \( y = a|x| \), where:

• If \( a > 0 \), the graph opens upward (V - shaped with the vertex at the bottom).
• If \( a < 0 \), the graph opens downward (V - shaped with the vertex at the top).

1. For \( A \):

The graph of \( y = A|x| \) opens upward (the vertex is the minimum point). So, \( A>0 \) (positive).

1. For \( B \):

The graph of \( y = B|x| \) opens upward (the vertex is the minimum point). So, \( B > 0\) (positive).

1. For \( C \):

The graph of \( y = C|x| \) opens downward (the vertex is the maximum point). So, \( C<0 \) (negative).

1. For \( D \):

The graph of \( y = D|x| \) opens downward (the vertex is the maximum point). So, \( D < 0\) (negative).

Part (b): Coefficient Closest to 0

The coefficient of \( |x| \) in \( y=a|x| \) determines the "steepness" of the graph. A coefficient with a smaller absolute value will make the graph less steep (closer to the \( x \) - axis).

• The graph of \( y = A|x| \) is the least steep among the upward - opening graphs (compared to \( y = B|x| \)) and also less steep than the downward - opening graphs when we consider the absolute value of the coefficients. So, the coefficient closest to 0 is \( A \).

Part (c): Coefficient with the Greatest Value

• For positive coefficients (\( A \) and \( B \)): The graph of \( y = B|x| \) is steeper than the graph of \( y = A|x| \), so \( B>A>0 \).
• For negative coefficients (\( C \) and \( D \)): Negative numbers are less than positive numbers. So, among all the coefficients, \( B \) (which is positive and has a larger absolute value than \( A \)) has the greatest value.

Answer:

s:

Part (a)

• \( A \): Positive
• \( B \): Positive
• \( C \): Negative
• \( D \): Negative

Part (b)

The coefficient closest to 0 is \( A \). So the answer is \( \boldsymbol{A} \).

Part (c)

The coefficient with the greatest value is \( B \). So the answer is \( \boldsymbol{B} \).