Question

identifying the characteristics of linear and exponential functions

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$$\begin{tabular}{|l|l|} \\hline function & type and behavior \\\\ \\hline \\(f(x) = 39x - 130\\) & select an answer select an answer \\\\ \\hline \\(p(t) = 230(1.15)^t\\) & select an answer select an answer \\\\ \\hline \\(d(y) = 6(2)^y\\) & select an answer select an answer \\\\ \\hline \\(h(x) = 70(0.7)^x\\) & select an answer select an answer \\\\ \\hline \\(g(x) = -17x + 150\\) & select an answer select an answer \\\\ \\hline \\end{tabular}$$

Explanation:

🆕 New Concept Discovered: Linear vs. Exponential Functions
Linear grows by adding; exponential grows by multiplying.

Step 1: Analyze \( f(x) = 39x - 130 \)

This function is in the slope-intercept form \( y = mx + b \), where \( m = 39 \) and \( b = -130 \).

• Type: Since the variable \( x \) is multiplied by a constant, it is a Linear function.
• Behavior: The slope \( m = 39 \) is positive, meaning the function increases by a constant amount for each unit increase in \( x \). Thus, it is Increasing.

Step 2: Analyze \( p(t) = 230(1.15)^t \)

This function is in the exponential form \( y = ab^t \), where \( a = 230 \) and the base \( b = 1.15 \).

• Type: Since the variable \( t \) is in the exponent, it is an Exponential function.
• Behavior: The base \( b = 1.15 \) is greater than \( 1 \), which means the value multiplies by \( 1.15 \) at each step. Thus, it is Increasing (Exponential Growth).

Step 3: Analyze \( d(y) = 6(2)^y \)

This function is in the exponential form \( ab^y \), where \( a = 6 \) and the base \( b = 2 \).

• Type: Since the variable \( y \) is in the exponent, it is an Exponential function.
• Behavior: The base \( b = 2 \) is greater than \( 1 \), meaning the value doubles at each step. Thus, it is Increasing (Exponential Growth).

Step 4: Analyze \( h(x) = 70(0.7)^x \)

This function is in the exponential form \( ab^x \), where \( a = 70 \) and the base \( b = 0.7 \).

• Type: Since the variable \( x \) is in the exponent, it is an Exponential function.
• Behavior: The base \( b = 0.7 \) is between \( 0 \) and \( 1 \), meaning the value shrinks to \( 70\% \) of its previous value at each step. Thus, it is Decreasing (Exponential Decay).

Step 5: Analyze \( g(x) = -17x + 150 \)

This function is in the slope-intercept form \( y = mx + b \), where \( m = -17 \) and \( b = 150 \).

• Type: Since the variable \( x \) is multiplied by a constant, it is a Linear function.
• Behavior: The slope \( m = -17 \) is negative, meaning the function decreases by \( 17 \) units for each unit increase in \( x \). Thus, it is Decreasing.

Answer:

Function	Type	Behavior
\( p(t) = 230(1.15)^t \)	Exponential	Increasing
\( d(y) = 6(2)^y \)	Exponential	Increasing
\( h(x) = 70(0.7)^x \)	Exponential	Decreasing
\( g(x) = -17x + 150 \)	Linear	Decreasing