Question

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

four graphs 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: (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 least value.
∘ a ∘ b ∘ c ∘ d

Explanation:

Step1: Analyze the graph of \( y = A|x| \)

The graph of \( y = A|x| \) opens upwards (since the V - shape is opening towards the positive y - direction). For the absolute - value function \( y = a|x| \), if the graph opens upwards, \( a>0 \). So \( A \) is positive.

Step2: Analyze the graph of \( y = B|x| \)

The graph of \( y = B|x| \) also opens upwards. So, similar to the analysis of \( A \), \( B>0 \).

Step3: Analyze the graph of \( y = C|x| \)

The graph of \( y = C|x| \) opens downwards (the V - shape is opening towards the negative y - direction). For the absolute - value function \( y = a|x| \), if the graph opens downwards, \( a < 0 \). So \( C \) is negative.

Step4: Analyze the graph of \( y = D|x| \)

The graph of \( y = D|x| \) opens downwards. So \( D<0 \).

Step5: Analyze the steepness (for part b and c)

The steepness of the graph of \( y=a|x| \) is determined by the absolute value of \( a \). The larger \( |a| \), the steeper the graph.
- For the upward - opening graphs (\( A \) and \( B \)): The graph of \( y = A|x| \) is steeper than the graph of \( y = B|x| \), so \( |A|>|B| \) and since \( A>0 \) and \( B>0 \), \( A > B>0 \).
- For the downward - opening graphs (\( C \) and \( D \)): The graph of \( y = C|x| \) is steeper than the graph of \( y = D|x| \), so \( |C|>|D| \) and since \( C<0 \) and \( D<0 \), \( C < D<0 \) (because for negative numbers, the larger the absolute value, the smaller the number. For example, \( - 5<-2 \) since \( | - 5|=5\) and \( | - 2| = 2\) and \( 5>2\)).

Part (b)

We want to find the coefficient closest to 0. We compare the absolute values of \( A,B,C,D \). Since \( |D| < |C| \) (because \( C \) is more negative, steeper) and \( |B|<|A| \) (because \( A \) is steeper). Also, we can see from the graphs that the graph of \( y = D|x| \) is the least steep among all the graphs (both upward and downward opening). So the coefficient closest to 0 is \( D \)? Wait, no. Wait, let's re - evaluate. Wait, the upward - opening graphs: \( B \) has a smaller absolute value than \( A \) (since it's less steep). The downward - opening graphs: \( D \) has a smaller absolute value than \( C \) (since it's less steep). Now we need to compare \( |B| \) and \( |D| \). Looking at the graphs, the graph of \( y = D|x| \) is a very shallow downward - opening graph and the graph of \( y = B|x| \) is a shallow upward - opening graph. But from the visual, the graph of \( y = D|x| \) seems to be the least steep, so \( |D| \) is the smallest, so \( D \) is closest to 0? Wait, no, wait. Wait, the y - intercept of all these graphs at \( x = 0 \) is 0. Let's take a point on each graph. For example, when \( x = 1 \):
- For \( y = A|x| \), \( y = A \); for \( y = B|x| \), \( y = B \); for \( y = C|x| \), \( y = C \); for \( y = D|x| \), \( y = D \).
Looking at the grids, the graph of \( y = D|x| \) at \( x = 1 \) is closer to 0 (has a value closer to 0) than the graph of \( y = B|x| \) at \( x = 1 \). And among the positive coefficients, \( B \) is smaller than \( A \), among the negative coefficients, \( D \) is larger than \( C \) (since \( D>-C\) in terms of value, but \( |D|<|C| \)). So the coefficient closest to 0 is \( D \)? Wait, no, wait. Wait, let's check the slopes. The slope of the right - hand side (for \( x>0 \)) of \( y = a|x| \) is \( a \) (for \( x>0 \), \( y = ax \)) and the slope of the left - hand side (for \( x < 0 \)) is \( - a \) (for \( x<0 \), \( y=-ax \)).
- For \( y = B|x| \) ( \( x>0 \), \( y = Bx \)): The slope \( B \) is small (shallow line).
- For \( y = D|x| \) ( \( x>0 \), \( y = Dx \)): The slope \( D \) is a small negative number (shallow line going down).
From the visual, the graph of \( y = D|x| \) is the least steep, so \( |D| \) is the smallest. So the coefficient closest to 0 is \( D \).

Part (c)

We want to find the coefficient with the least value. We know that \( C \) and \( D \) are negative, \( A \) and \( B \) are positive. Negative numbers are less than positive numbers. Among the negative numbers, \( C < D \) (because \( |C|>|D| \), and for negative numbers, the larger the absolute value, the smaller the number. For example, if \( C=-3 \) and \( D = - 1 \), then \( - 3<-1\)). Among the positive numbers, \( B < A \). So the least value is \( C \) (since \( C \) is a negative number with a larger absolute value than \( D \), so it's smaller than \( D \), and negative numbers are smaller than positive numbers).

Part (a)

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

Part (b)

The coefficient closest to 0 is \( D \) (because \( |D| \) is the smallest among \( |A|,|B|,|C|,|D| \))

Part (c)

The coefficient with the least value is \( C \) (because \( C \) is a negative number with a larger absolute value than \( D \), so \( C < D<0 \) and \( 0 < B < A \), so \( C \) is the smallest)

Answer:

(a)

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

(b) \(\boldsymbol{D}\)

(c) \(\boldsymbol{C}\)