Question

compare the tables for situations w and z. use the patterns in the tables to complete the statements. select the correct answer from each drop-down menu. situation w represents a linear relationship because it has a common difference of 5. situation z represents an exponential relationship because it has a common ratio of 3. part c use what youve found out about the relationship between the variables in each situation to model them using other methods. question write equations for situation w and situation z. then graph each equation, and use the mark button to label each

Explanation:

Step1: Analyze Situation W (Linear Relationship)

A linear relationship has the form \( y = mx + b \), where \( m \) is the slope (common difference). If the common difference is 5, let's assume the initial value (when \( x = 0 \)) is \( b \). For a linear sequence, the equation is \( y = 5x + b \). If we take the first term (e.g., when \( x = 1 \), \( y = 5(1)+b \)), but since we know the common difference is 5, the general linear equation for Situation W is \( y = 5x + b \) (we can determine \( b \) if we have a point, but the key is the slope is 5).

Step2: Analyze Situation Z (Exponential Relationship)

An exponential relationship has the form \( y = ab^x \), where \( b \) is the common ratio. Given the common ratio is 3, the equation is \( y = a(3)^x \), where \( a \) is the initial value (when \( x = 0 \), \( y = a \)).

Step3: Write Equations

• Situation W (Linear): Let's assume the initial value (when \( x = 0 \)) is \( y_0 \). Then the equation is \( y = 5x + y_0 \). If we take a sample point (e.g., if when \( x = 0 \), \( y = y_0 \), and when \( x = 1 \), \( y = y_0 + 5 \)), the linear equation is \( y = 5x + y_0 \). For simplicity, if the initial value is 0 (or any value), the slope is 5, so the equation is \( y = 5x + c \) (where \( c \) is the y-intercept).
• Situation Z (Exponential): Let the initial value (when \( x = 0 \)) be \( a \). Then the equation is \( y = a(3)^x \). For example, if \( a = 1 \), the equation is \( y = 3^x \); if \( a \) is another value, it's \( y = a(3)^x \).

Answer:

• Equation for Situation W (Linear): \( y = 5x + b \) (where \( b \) is the y - intercept; if we assume \( b = 0 \) for a simple case, \( y = 5x \))
• Equation for Situation Z (Exponential): \( y = a(3)^x \) (where \( a \) is the initial value; if we assume \( a = 1 \) for a simple case, \( y = 3^x \))

(Note: To graph \( y = 5x \) (linear), it's a straight line with slope 5 passing through the origin. To graph \( y = 3^x \) (exponential), it's a curve that passes through \( (0,1) \) and increases rapidly, with a horizontal asymptote at \( y = 0 \).)