To graph the function [tex]\( g(x) = \frac{5}{8}(x+4)^2 - 3 \)[/tex] based on the original function [tex]\( f(x) = x^2 \)[/tex], we need to apply a sequence of transformations. Here are the step-by-step transformations required:
1. Horizontal Shift Left by 4 Units:
- Start with the original function [tex]\( f(x) = x^2 \)[/tex].
- To shift the graph horizontally to the left by 4 units, we replace [tex]\( x \)[/tex] with [tex]\( x + 4 \)[/tex]. The new function becomes [tex]\( f(x + 4) = (x + 4)^2 \)[/tex].
2. Vertical Stretch by a Factor of [tex]\(\frac{5}{8}\)[/tex]:
- Now take the function [tex]\( (x + 4)^2 \)[/tex].
- To apply a vertical stretch by a factor of [tex]\(\frac{5}{8}\)[/tex], we multiply the entire function by [tex]\(\frac{5}{8}\)[/tex]. The resulting function is [tex]\( \frac{5}{8} (x + 4)^2 \)[/tex].
3. Vertical Shift Down by 3 Units:
- Finally, take the vertically stretched function [tex]\( \frac{5}{8} (x + 4)^2 \)[/tex].
- To shift the graph vertically downward by 3 units, we subtract 3 from the entire function. The final function becomes [tex]\( \frac{5}{8} (x + 4)^2 - 3 \)[/tex].
To summarize, the sequence of transformations required to graph [tex]\( g(x) = \frac{5}{8}(x+4)^2 - 3 \)[/tex] based on [tex]\( f(x) = x^2 \)[/tex] are:
1. Horizontal shift left by 4 units.
2. Vertical stretch by a factor of [tex]\(\frac{5}{8}\)[/tex].
3. Vertical shift down by 3 units.
