Let's analyze the transformation [tex]\( g(x) = 5 f(x) \)[/tex] step-by-step to understand its effects on the original function [tex]\( f(x) \)[/tex].
### 1. Understanding the Transformation
The transformation [tex]\( g(x) = 5 f(x) \)[/tex] can be described as a vertical stretch of the original function [tex]\( f(x) \)[/tex] by a factor of 5. This means that for any given value of [tex]\( x \)[/tex], the output of [tex]\( f(x) \)[/tex] is multiplied by 5 to get [tex]\( g(x) \)[/tex].
### 2. Effects on the Slope
If the original function [tex]\( f(x) \)[/tex] is linear and can be expressed in the form [tex]\( f(x) = mx + b \)[/tex]:
- The slope of [tex]\( f(x) \)[/tex] is [tex]\( m \)[/tex].
- For the transformed function [tex]\( g(x) = 5 f(x) \)[/tex], this becomes [tex]\( g(x) = 5(mx + b) = 5mx + 5b \)[/tex].
So, the new slope becomes [tex]\( 5m \)[/tex], which means the slope is increased by a factor of 5. It does not change signs.
### 3. Effects on the Domain
The domain of the function [tex]\( f(x) \)[/tex] refers to all the input values [tex]\( x \)[/tex] for which the function is defined. Since the transformation [tex]\( g(x) = 5 f(x) \)[/tex] only involves scaling the output by 5 and does not change how [tex]\( x \)[/tex] is processed, the domain of the function remains unaffected. Therefore, the domain does not change.
### 4. Effects on the [tex]\( x \)[/tex]-Intercept
The [tex]\( x \)[/tex]-intercept is the point where the function crosses the [tex]\( x \)[/tex]-axis. For the original function [tex]\( f(x) \)[/tex]:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex].
For the transformed function [tex]\( g(x) = 5 f(x) \)[/tex], the [tex]\( x \)[/tex]-intercept occurs where [tex]\( 5 f(x) = 0 \)[/tex]. Since multiplying zero by 5 still gives zero, the [tex]\( x \)[/tex]-intercept remains unchanged. Therefore, the [tex]\( x \)[/tex]-intercept does not increase by a multiple of 5.
### 5. Effects on the [tex]\( y \)[/tex]-Intercept
The [tex]\( y \)[/tex]-intercept is the point where the function crosses the [tex]\( y \)[/tex]-axis, i.e., where [tex]\( x = 0 \)[/tex].
For the original function [tex]\( f(x) \)[/tex]:
- The [tex]\( y \)[/tex]-intercept is [tex]\( f(0) = b \)[/tex], where [tex]\( b \)[/tex] is the constant term.
For the transformed function [tex]\( g(x) = 5 f(x) \)[/tex]:
- The [tex]\( y \)[/tex]-intercept is [tex]\( g(0) = 5 f(0) = 5b \)[/tex].
Therefore, if the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is non-zero, it increases by a factor of 5.
### Conclusion
Based on our analysis, the correct effect of the transformation [tex]\( g(x) = 5 f(x) \)[/tex] on the original function [tex]\( f(x) \)[/tex] is:
- The [tex]\( y \)[/tex]-intercept increased by a multiple of 5.
Hence, the answer is:
The [tex]\( y \)[/tex]-intercept increased by a multiple of 5.
