Question

(a) for each coefficient, choose whether it is positive or negative.
a: positive b: positive c: negative d: negative
(b) choose the coefficient closest to 0.
a b c d
(c) choose the coefficient with the least value.
a b c d

Explanation:

Part (b)

The general form of an absolute - value function is \(y = a|x|\). The magnitude of \(|a|\) determines the "steepness" of the graph. A smaller magnitude of \(a\) (closer to 0) means the graph is flatter (less steep).

• For \(y = A|x|\), the graph is very steep (narrow), so \(|A|\) is large.
• For \(y = B|x|\), the graph is relatively flat compared to \(y = A|x|\), \(y = C|x|\), and \(y = D|x|\) (when we consider the steepness).
• For \(y = C|x|\) and \(y = D|x|\), the graphs are steeper than \(y = B|x|\) (they have a larger slope in the non - vertex regions compared to \(y = B|x|\)).

So, the coefficient with the magnitude closest to 0 is \(B\) because its graph is the flattest among the four.

The function is of the form \(y=a|x|\). The value of \(a\) determines the direction (positive or negative) and the steepness.

• For \(A\) and \(B\), \(a\) is positive. For \(C\) and \(D\), \(a\) is negative. Negative numbers are less than positive numbers.
• Now, we compare the magnitudes of \(C\) and \(D\). The graph of \(y = C|x|\) is steeper than the graph of \(y = D|x|\). The magnitude of the coefficient of \(y = C|x|\) (i.e., \(|C|\)) is larger than the magnitude of the coefficient of \(y = D|x|\) (i.e., \(|D|\)). Since \(C\) and \(D\) are negative, a larger magnitude of a negative number means the number is smaller. For example, if \(C=-2\) and \(D = - 1\), then \(-2<-1\). So, among \(A\) (positive), \(B\) (positive), \(C\) (negative), and \(D\) (negative), \(C\) has the least value.

Answer:

B

Part (c)