Question

analyze the functions c(x) and b(x) to determine which has fewer zeros. use the two representations, found below, to compare the two relationships. a linear function, c(x), that crosses the y - axis at (0, -11) and the x - axis at (-11, 0) with a rate of change of -1. b(x) how many zeros does this function have? function b(x) has how many zeros? which function has fewer zeros, including any imaginary zeros and multiplicity?

Explanation:

Step1: Analyze function \( c(x) \)

\( c(x) \) is a linear function. A linear function \( y = mx + b \) has at most 1 zero (where \( y = 0 \)). Given \( c(x) \) crosses the \( x \)-axis at \( (-11, 0) \), so \( c(x) \) has 1 zero.

Step2: Analyze function \( b(x) \)

From the graph of \( b(x) \), we identify the \( x \)-intercepts (zeros) by looking at where the graph crosses the \( x \)-axis. The points are \( (-2, 0) \), \( (0, 0) \), and we check the graph's behavior. Wait, no—wait, the graph of \( b(x) \): let's re - examine. Wait, the graph of \( b(x) \) crosses the \( x \)-axis at \( x=-2 \), \( x = 0 \), and does it cross again? Wait, no, looking at the graph: the points marked are \( (-2,0) \), \( (0,0) \)? Wait, no, maybe I misread. Wait, the graph of \( b(x) \): let's see the \( x \)-intercepts. The graph crosses the \( x \)-axis at \( x=-2 \), \( x = 0 \), and is there a third? Wait, no, the graph as shown: the curve crosses the \( x \)-axis at \( (-2, 0) \) and \( (0, 0) \)? Wait, no, maybe the multiplicity? Wait, no, a cubic - like graph? Wait, no, the key is: linear function \( c(x) \) has 1 zero. For \( b(x) \), from the graph, how many times does it cross the \( x \)-axis? Let's count the \( x \)-intercepts. The graph of \( b(x) \) crosses the \( x \)-axis at \( x=-2 \), \( x = 0 \), and maybe another? Wait, no, looking at the graph: the points are \( (-2,0) \), \( (0,0) \), and when \( x = 1.8 \), \( y=0 \)? Wait, no, the label \( (1.8, 0) \)? Wait, the graph has points: \( (-2,0) \), \( (0,0) \), and \( (1.8, 0) \)? Wait, no, the user's graph: let's re - interpret. Wait, the graph of \( b(x) \): the \( x \)-intercepts are at \( x=-2 \), \( x = 0 \), and \( x=1.8 \)? Wait, no, maybe I made a mistake. Wait, no—wait, the linear function \( c(x) \) has 1 zero. For \( b(x) \), from the graph, let's count the number of times it intersects the \( x \)-axis. The graph crosses the \( x \)-axis at three points? Wait, no, the graph shown: the curve goes through \( (-2,0) \), \( (0,0) \), and \( (1.8, 0) \)? Wait, no, maybe the graph is a cubic function? Wait, no, the key is: linear function \( c(x) \) has 1 zero. For \( b(x) \), how many zeros? Let's check again. The graph of \( b(x) \): the \( x \)-intercepts are \( x=-2 \), \( x = 0 \), and \( x = 1.8 \)? Wait, no, the label \( (1.8, 0) \) is on the \( x \)-axis? Wait, the point \( (1.8, 0) \) is on the \( x \)-axis, so that's a zero. Wait, so \( b(x) \) has 3 zeros? Wait, no, maybe I messed up. Wait, no, the linear function \( c(x) \): \( y=mx + b \), with \( m=-1 \), \( b=-11 \), so \( c(x)=-x - 11 \). Setting \( y = 0 \), \( -x-11=0\Rightarrow x=-11 \), so 1 zero. For \( b(x) \), from the graph, the number of \( x \)-intercepts (zeros) is 3? Wait, no, the graph as shown: the curve crosses the \( x \)-axis at \( (-2, 0) \), \( (0, 0) \), and \( (1.8, 0) \)? Wait, no, the point \( (0,0) \): is that a zero? Wait, when \( x = 0 \), \( y = 0 \), so that's a zero. So \( b(x) \) has 3 zeros? Wait, no, maybe the graph is of a function that crosses the \( x \)-axis at three points? Wait, no, the linear function \( c(x) \) has 1 zero, and \( b(x) \) has more than 1. Wait, maybe I misread the graph. Wait, the graph of \( b(x) \): let's see the points. The graph has a point at \( (-2, 0) \), \( (0, 0) \), and \( (1.8, 0) \)? So three zeros. But wait, the linear function \( c(x) \) has 1 zero. So \( c(x) \) has fewer zeros.

Answer:

Function \( c(x) \) has fewer zeros.