Question

question 26
starting with the graph of ( f(x)=6^{x} ), write the equation of the graph that results when:
(a) ( f(x) ) is shifted 6 units upward. ( y= )
(b) ( f(x) ) is shifted 2 units to the left. ( y= )
(c) ( f(x) ) is reflected about the ( x )-axis. ( y= )
question help: video

Explanation:

Step1: Vertical shift formula

For a function \(y = f(x)\), shifting \(k\) units upward gives \(y=f(x)+k\). Here \(f(x)=6^{x}\) and \(k = 6\), so \(y=6^{x}+6\).

Step2: Horizontal shift formula

For a function \(y = f(x)\), shifting \(h\) units to the left gives \(y = f(x + h)\). Here \(f(x)=6^{x}\) and \(h=2\), so \(y=6^{x + 2}\).

Step3: Reflection about the x - axis formula

For a function \(y = f(x)\), reflection about the \(x\) - axis gives \(y=-f(x)\). Here \(f(x)=6^{x}\), so \(y=-6^{x}\).

Answer:

(a) \(y = 6^{x}+6\)
(b) \(y = 6^{x + 2}\)
(c) \(y=-6^{x}\)