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Starting with the graph of [tex]$f(x)=3^x$[/tex], write the equation of the graph that results when:

(a) [tex]$f(x)$[/tex] is shifted 1 unit upward. [tex]y = \boxed{ }[/tex]

(b) [tex][tex]$f(x)$[/tex][/tex] is shifted 3 units to the right. [tex]y = \boxed{ }[/tex]

(c) [tex]$f(x)$[/tex] is reflected about the [tex]$y$[/tex]-axis. [tex]y = \boxed{ }[/tex]

Sagot :

GinnyAnsweridn
Certainly! Let's transform the function [tex]\( f(x) = 3^x \)[/tex] as indicated:

### (a) Upward shift of 1 unit:
When you shift the graph of a function [tex]\( f(x) \)[/tex] upward by [tex]\( k \)[/tex] units, you add [tex]\( k \)[/tex] to the function. Therefore, if we shift [tex]\( f(x) = 3^x \)[/tex] upward by 1 unit, the new function becomes:
[tex]\[ y = 3^x + 1 \][/tex]

### (b) Right shift of 3 units:
When you shift the graph of a function [tex]\( f(x) \)[/tex] to the right by [tex]\( h \)[/tex] units, you replace [tex]\( x \)[/tex] with [tex]\( x - h \)[/tex]. Therefore, if we shift [tex]\( f(x) = 3^x \)[/tex] to the right by 3 units, the new function becomes:
[tex]\[ y = 3^{(x - 3)} \][/tex]

### (c) Reflection about the y-axis:
When you reflect the graph of a function [tex]\( f(x) \)[/tex] about the y-axis, you replace [tex]\( x \)[/tex] with [tex]\( -x \)[/tex]. Therefore, the reflection of [tex]\( f(x) = 3^x \)[/tex] about the y-axis gives us the new function:
[tex]\[ y = 3^{-x} \][/tex]

Thus, the equations for the given transformations are:

- (a) [tex]\( y = 3^x + 1 \)[/tex]
- (b) [tex]\( y = 3^{(x-3)} \)[/tex]
- (c) [tex]\( y = 3^{-x} \)[/tex]
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