Questions: Graph the function by using transformations of the graph of y=1/x^2. Plot all necessary asymptotes. For vertical asymptotes, make sure there are at least two points on each side. g(x)=1/x^2-5

Graph the function by using transformations of the graph of y=1/x^2. Plot all necessary asymptotes. For vertical asymptotes, make sure there are at least two points on each side.

g(x)=1/x^2-5

Transcript text: Graph the function by using transformations of the graph of $y=\frac{1}{x^{2}}$. Plot all necessary asymptotes. For vertical asymptotes, make sure there are at least two points on each side. \[ g(x)=\frac{1}{x^{2}}-5 \]

Solution

Solution Steps

Step 1: Identify the given function

The given function is: \[ g(x) = \frac{1}{x^2} - 5 \]

Step 2: Determine the transformations

The function $ g(x) $ is a transformation of $ y = \frac{1}{x^2} $. Specifically, it is shifted downward by 5 units.

Step 3: Identify the asymptotes

The vertical asymptote is at $ x = 0 $ because the function $ \frac{1}{x^2} $ has a vertical asymptote at $ x = 0 $. The horizontal asymptote is at $ y = -5 $ because the function $ \frac{1}{x^2} $ approaches 0 as $ x $ approaches infinity, and shifting it down by 5 units moves the horizontal asymptote to $ y = -5 $.

Final Answer

The function $ g(x) = \frac{1}{x^2} - 5 $ has a vertical asymptote at $ x = 0 $ and a horizontal asymptote at $ y = -5 $.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = (1/x**2) - 5"], "latex_expressions": ["$g(x) = \\frac{1}{x^2} - 5$"]}

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