Question

starting with the graph of (f(x) = 8^x), write the equation of the graph that results from

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

b. shifting (f(x)) 5 units right.
(y = )

c. shifting (f(x)) 6 units upward.
(y = )

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Reflection about the x-axis

A reflection across the \(x\)-axis multiplies the entire function by \(-1\):

$$ y = -f(x) $$

Applying this to \(f(x) = 8^x\):

$$ y = -8^x $$

Step 2: Horizontal shift right

A horizontal shift to the right by \(h\) units replaces \(x\) with \(x - h\):

$$ y = f(x - h) $$

For a shift of \(5\) units to the right:

$$ y = 8^{x - 5} $$

Step 3: Vertical shift upward

A vertical shift upward by \(k\) units adds \(k\) to the entire function:

$$ y = f(x) + k $$

For a shift of \(6\) units upward:

$$ y = 8^x + 6 $$

Answer:

a. \(y = -8^x\)

b. \(y = 8^{x - 5}\)

c. \(y = 8^x + 6\)