To graph the function [tex]\( h(x) = -4x^5 \)[/tex] using a transformation of the graph of [tex]\( y = x^5 \)[/tex], we can follow a few steps:
1. Start with the basic graph:
The graph of [tex]\( y = x^5 \)[/tex] is a well-known cubic-like curve. For positive [tex]\( x \)[/tex], the values of [tex]\( y \)[/tex] increase rapidly since [tex]\( x^5 \)[/tex] becomes very large as [tex]\( x \)[/tex] increases. For negative [tex]\( x \)[/tex], the values of [tex]\( y \)[/tex] decrease rapidly because [tex]\( x^5 \)[/tex] becomes very large in the negative direction.
2. Vertical stretch:
The term [tex]\( -4 \)[/tex] in [tex]\( h(x) = -4x^5 \)[/tex] indicates a vertical stretch. Specifically, every [tex]\( y \)[/tex]-value of the graph [tex]\( y = x^5 \)[/tex] is multiplied by 4. This means any point [tex]\( (x, x^5) \)[/tex] on the graph of [tex]\( y = x^5 \)[/tex] will be transformed to [tex]\( (x, 4x^5) \)[/tex].
3. Reflection across the x-axis:
The negative sign before the 4 indicates that the graph of the function [tex]\( y = x^5 \)[/tex] will be reflected across the x-axis. After reflection, the point [tex]\( (x, 4x^5) \)[/tex] becomes [tex]\( (x, -4x^5) \)[/tex].
4. Graph the new function:
The final function [tex]\( h(x) = -4x^5 \)[/tex] will therefore be a vertically stretched (by a factor of 4) and reflected (across the x-axis) version of [tex]\( y = x^5 \)[/tex].
Visual interpretation:
- At small values of [tex]\( x \)[/tex], the curve [tex]\( h(x) = -4x^5 \)[/tex] will appear similar to [tex]\( y = x^5 \)[/tex] but will be flipped upside-down and stretched, making it steeper.
- At large values of [tex]\( x \)[/tex], the curve will descend rapidly because of the '-4' factor.
Since we're choosing between multiple graph options labeled A, B, C, and D, you will need to look for the graph that matches these transformations of the function [tex]\( y = x^5 \)[/tex]:
- It should involve a reflection across the x-axis (so it curves downward for positive [tex]\( x \)[/tex]).
- It should be stretched vertically by a factor of 4 (so it is steeper than the basic [tex]\( y = x^5 \)[/tex] curve).
Given these transformations, identify the graph in the choices that satisfies these conditions. The correct graph will reflect the features we described:
- Steeper curve
- Reflected across the x-axis
After careful comparison, select the one that accurately matches these transformations.
