Question

1. write an exponential function with a y - intercept of 7 and a decay rate of 75%. type a response 10. the parent graph ( f(x) = 2^x ) is translated to the right 5 units and down 3 units, along with a reflection across the x - axis. write the transformed function. type a response

Explanation:

Problem 9:

Step 1: Recall the exponential function form

The general form of an exponential function is \( y = a(b)^x \), where \( a \) is the initial value (y - intercept) and \( b \) is the base. For decay, \( 0 < b < 1 \). The decay rate \( r = 75\% = 0.75 \), so the base \( b = 1 - r = 1 - 0.75 = 0.25 \). The y - intercept \( a = 7 \).

Step 2: Write the function

Substitute \( a = 7 \) and \( b = 0.25 \) into the general form. So the exponential function is \( y = 7(0.25)^x \).

Step 1: Recall transformation rules

• Reflection across the x - axis: If we have a function \( y = f(x) \), its reflection across the x - axis is \( y=-f(x) \).
• Horizontal shift: If we shift a function \( y = f(x) \) to the right by \( h \) units, the function becomes \( y = f(x - h) \).
• Vertical shift: If we shift a function \( y = f(x) \) down by \( k \) units, the function becomes \( y = f(x)-k \).

Step 2: Apply transformations to \( f(x)=2^x \)

• First, reflection across the x - axis: \( y=-2^x \).
• Then, shift right 5 units: Replace \( x \) with \( x - 5 \), so \( y=-2^{x - 5} \).
• Then, shift down 3 units: Subtract 3 from the function, so \( y=-2^{x - 5}-3 \).

Answer:

\( y = 7(0.25)^x \)

Problem 10: