Use the graph of the parent function, f left-parenthesis x right-parenthesis equals Start Root x End Root, to describe the transformation for the graph of g left-parenthesis x right-parenthesis equals Start Root x minus 5 End Root.
The graph of the parent function f(x) = √x is a simple square root function. The graph starts at the point (0, 0) and has a horizontal asymptote at y = 0. As x increases, y also increases but at a decreasing rate.
The transformation g(x) = √(x - 5) shifts the graph of the parent function 5 units to the right. This means the graph will now start at the point (5, 0) instead of (0, 0). The horizontal asymptote remains at y = 0, and as x increases, y also increases but at a slower rate compared to the graph of f(x).
In summary, the graph of g(x) = √(x - 5) is a shifted and slightly compressed version of the graph of f(x) = √x.
Which function transforms the graph of f left-parenthesis x right-parenthesis equals x superscript 5 baseline by shifting it up 6 units?
options
a) g left-parenthesis x right-parenthesis equals x superscript 5 baseline plus 6Image with alt text: g left-parenthesis x right-parenthesis equals x superscript 5 baseline plus 6
b) g left-parenthesis x right-parenthesis equals x superscript 5 baseline minus 6
Image with alt text: g left-parenthesis x right-parenthesis equals x superscript 5 baseline minus 6
c) g left-parenthesis x right-parenthesis equals left-parenthesis x plus 6 right-parenthesis superscript 5 baseline
Image with alt text: g left-parenthesis x right-parenthesis equals left-parenthesis x plus 6 right-parenthesis superscript 5 baseline
d) g left-parenthesis x right-parenthesis equals left-parenthesis x minus 6 right-parenthesis superscript 5 baseline
The correct option is:
c) g(x) = (x + 6)^5
To describe the transformation of the graph from the parent function f(x) = √x to the function g(x) = √(x - 5), we need to consider the effect of the subtraction of 5 inside the square root. Here are the steps of the transformation:
1. Horizontal shift: The function g(x) = √(x - 5) is obtained by shifting the graph 5 units to the right compared to the parent function f(x) = √x.
2. New x-intercept: The parent function f(x) = √x has an x-intercept at x = 0. However, for the function g(x) = √(x - 5), the new x-intercept occurs when (x - 5) = 0, which implies x = 5. So, the new x-intercept is at x = 5.
3. Vertical shift: There is no vertical shift involved in this transformation since there is no addition or subtraction outside the square root.
Overall, the transformation for the graph of g(x) = √(x - 5) compared to the parent function f(x) = √x involves a horizontal shift of 5 units to the right and a new x-intercept at x = 5.