Question

y = 2·(\frac{6}{7})^x
c. exponential growth
e. neither
d. exponential decay
f. both

1. if your parent function is g(x) = 2^x, how could you get the graph of

f(x) = -2^x + 4 ?
a. move g(x) up 4 then reflect across the y - axis to get f(x).
b. move g(x) down 4 then reflect across the x - axis to get f(x).
c. move g(x) right 4 then reflect across the x - axis to get f(x).
d. move g(x) left 4 then reflect across the y - axis to get f(x).

1. compare the exponential functions f(x) = 8(0.75)^x and g(x) = 8(1.25)^x. which statement is incorrect?

a) both functions have a horizontal asymptote at y = 0 and a y - intercept at (0,8).
b) each time x increases by 1, f(x) decreases by 25% and g(x) increases by 25%.
c) both functions have range y > 0.
d) both functions are increasing.

Explanation:

Question 5

Step1: Analyze Transformations

The parent function is \( g(x) = 2^x \). The function \( f(x)=-2^x + 4 \) involves two transformations: reflection and vertical shift. First, the negative sign in front of \( 2^x \) reflects \( g(x) \) across the \( x \)-axis (since reflecting \( y = a^x \) across \( x \)-axis gives \( y=-a^x \)). Then, adding 4 shifts the graph up 4 units? Wait, no, wait: Wait, \( f(x)=-2^x + 4 \) is \( -g(x)+4 \). Wait, the options: Let's check each option.

Option a: Move up 4 then reflect across y-axis. Reflect across y-axis would be \( 2^{-x} \), then move up 4: \( 2^{-x}+4 \), not \( -2^x +4 \). So a is wrong.

Option b: Move down 4 then reflect across x-axis. Wait, no: Wait, \( f(x)=-2^x + 4 = -g(x)+4 \). So first, reflect \( g(x) \) across x-axis: \( -g(x)=-2^x \), then shift up 4 units: \( -2^x +4 \). Wait, but the option b says "move down 4 then reflect across x-axis". Wait, maybe I misread. Wait, let's re-express \( f(x) \): \( f(x)= -2^x + 4 = 4 - 2^x \). Alternatively, think of order: reflection over x-axis (changes \( 2^x \) to \( -2^x \)) then vertical shift up 4? But the options: Let's check each option again.

Option a: Move up 4 (so \( 2^x +4 \)) then reflect across y-axis (so \( 2^{-x}+4 \)) → not \( -2^x +4 \). So a is wrong.

Option b: Move down 4 ( \( 2^x -4 \)) then reflect across x-axis ( \( -(2^x -4)= -2^x +4 \)) → that's correct! Wait, let's verify: If we take \( g(x)=2^x \), move down 4: \( 2^x -4 \), then reflect across x-axis: \( -(2^x -4)= -2^x +4 \), which is \( f(x) \). So option b is correct? Wait, but let's check other options.

Option c: Move right 4: \( 2^{x - 4} \), then reflect across x-axis: \( -2^{x - 4} \), not \( -2^x +4 \). So c is wrong.

Option d: Move left 4: \( 2^{x + 4} \), reflect across y-axis: \( 2^{-x - 4} \), not \( -2^x +4 \). So d is wrong. So the correct option is b.

Step2: Confirm Transformations

Reflecting across x-axis changes \( y = h(x) \) to \( y=-h(x) \). Shifting down 4 units changes \( g(x) \) to \( g(x)-4 \). So \( -(g(x)-4)= -g(x)+4 \), which is \( f(x) \). So the correct transformation is move down 4 then reflect across x-axis, which is option b.

Step1: Analyze Exponential Functions

For \( f(x)=8(0.75)^x \): The base \( 0.75 < 1 \), so it's an exponential decay function (decreasing as \( x \) increases). For \( g(x)=8(1.25)^x \): The base \( 1.25 > 1 \), so it's an exponential growth function (increasing as \( x \) increases).

Now check each option:

a) Horizontal asymptote: For exponential functions \( y = ab^x \), horizontal asymptote is \( y = 0 \) (since as \( x \to \pm\infty \), \( b^x \to 0 \) if \( |b|
eq 1 \)). Y-intercept: when \( x = 0 \), \( f(0)=8(0.75)^0 = 8(1)=8 \), \( g(0)=8(1.25)^0 = 8(1)=8 \). So y-intercept at (0,8). So a is correct.

b) For \( f(x) \): When \( x \) increases by 1, \( f(x + 1)=8(0.75)^{x + 1}=8(0.75)^x(0.75) \). So it's 75% of the previous value, which means it decreases by 25% (since \( 1 - 0.75 = 0.25 \)). For \( g(x) \): \( g(x + 1)=8(1.25)^{x + 1}=8(1.25)^x(1.25) \), which is 125% of the previous value, so it increases by 25% (since \( 1.25 - 1 = 0.25 \)). So b is correct.

c) Range: For exponential functions \( y = ab^x \) with \( a > 0 \) and \( b > 0, b
eq 1 \), the range is \( y > 0 \) (since \( b^x > 0 \) for all real \( x \), so \( ab^x > 0 \)). So both have range \( y > 0 \). So c is correct.

d) \( f(x) \) has base \( 0.75 < 1 \), so it's decreasing. \( g(x) \) has base \( 1.25 > 1 \), so it's increasing. So the statement "Both functions are increasing" is incorrect.

Step2: Identify Incorrect Statement

Option d says both are increasing, but \( f(x) \) is decreasing. So d is the incorrect statement.

Answer:

b. Move \( g(x) \) down 4 then reflect across the x-axis to get \( f(x) \).

Question 6