Question

starting with the graph of (f(x) = 7^x), write the equation of the graph that results from

a. shifting (f(x)) 4 units left.
(y = )

b. reflecting (f(x)) about the x-axis and the y-axis.
(y = )

c. shifting (f(x)) 3 units downward.
(y = )

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Shift left by 4 units

To shift a function \( f(x) \) horizontally to the left by \( c \) units, replace \( x \) with \( x + c \):

$$ y = f(x + 4) = 7^{x + 4} $$

Step 2: Reflect about the x-axis and y-axis

To reflect a function \( f(x) \) about the x-axis, multiply the entire function by \(-1\): \( -f(x) \).
To reflect about the y-axis, replace \( x \) with \( -x \): \( f(-x) \).
Combining both reflections:

$$ y = -f(-x) = -7^{-x} $$

Step 3: Shift downward by 3 units

To shift a function \( f(x) \) vertically downward by \( d \) units, subtract \( d \) from the entire function:

$$ y = f(x) - 3 = 7^x - 3 $$

Answer:

a. \( y = 7^{x + 4} \)

b. \( y = -7^{-x} \)

c. \( y = 7^x - 3 \)