Question

look at the graphs and their equations below. then fill in the information about the coefficients a, b, c, and d.
(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 least value.
a b c d
(c) choose the coefficient closest to 0.
a b c d

Explanation:

Step1: Analyze the form of the function

The general form of the absolute value function is \( y = k|x| \), where \( k \) is the coefficient. The sign of \( k \) determines the direction (positive \( k \) opens upwards, negative \( k \) opens downwards), and the magnitude of \( |k| \) determines the "steepness" (larger \( |k| \) means steeper, smaller \( |k| \) means flatter, closer to 0).

Step2: Determine the sign of each coefficient (part a)

• For \( y = A|x| \): The graph opens upwards (since the V - shape is above the x - axis in the positive x - direction and symmetric in a way that the slope is positive in the right half), so \( A \) is positive.
• For \( y = B|x| \): The graph opens upwards, so \( B \) is positive.
• For \( y = C|x| \): The graph opens downwards (the V - shape is going downwards as \( x \) moves away from 0 in both directions), so \( C \) is negative.
• For \( y = D|x| \): The graph opens downwards, so \( D \) is negative.

Step3: Determine the coefficient with the least value (part b)

The value of a negative number is less than a positive number. Among \( C \) and \( D \), we look at the steepness. A more negative coefficient (with a larger magnitude) will have a steeper graph. If \( D \) is less steep (closer to horizontal) than \( C \), then \( D \) is greater than \( C \) (since \( C \) and \( D \) are negative, a less negative number is greater). Wait, no: for negative numbers, the one with the larger magnitude is smaller. For example, \( - 2<-1 \). So if \( C \) is steeper (more negative) than \( D \), then \( C \) has a smaller value. But let's assume from the graphs: usually, the steeper the negative slope, the more negative the coefficient. If \( C \) is steeper (more negative) than \( D \), then \( C \) is the least. But maybe from the given graphs, let's re - check. Wait, the first two graphs (\( A \) and \( B \)) open up (positive coefficients), the last two (\( C \) and \( D \)) open down (negative coefficients). To find the least value (most negative), we need the one with the largest magnitude of the negative coefficient. If \( C \) is steeper (more negative) than \( D \), then \( C \) is the least.

Step4: Determine the coefficient closest to 0 (part c)

The coefficient closest to 0 is the one with the smallest magnitude (the flattest graph). Among all four, the graph of \( A|x| \) seems to be the flattest (least steep) among the positive ones, and if we compare with the negative ones, the positive ones are above 0, and the flattest positive graph (smallest positive magnitude) or the flattest negative graph (smallest negative magnitude). Wait, the graph of \( A|x| \) looks the flattest (least steep), so \( A \) has the smallest magnitude (closest to 0) among the positive coefficients, and since positive numbers are greater than 0 and negative numbers are less than 0, the flattest graph (smallest \( |k| \)) will have \( k \) closest to 0. If \( A \) is the flattest, then \( A \) is closest to 0.

Answer:

(a) \( A \): Positive, \( B \): Positive, \( C \): Negative, \( D \): Negative
(b) The coefficient with the least value is \( C \) (assuming \( C \) is more negative than \( D \))
(c) The coefficient closest to 0 is \( A \) (assuming \( A \) is the flattest graph)