Question

231c1_25-26 - c - rusd math 3 semester a_90-89776-9332-1231c1
graphing rational functions
graphing multiples of a parent function
which is the graph of $g(x) = \frac{10}{x}$?
three image placeholders for graphs

Explanation:

Step1: Recall Parent Function

The parent function of \( g(x) = \frac{10}{x} \) is \( f(x)=\frac{1}{x} \), which is a hyperbola with two branches in the first and third quadrants (since the coefficient of \( \frac{1}{x} \) is positive).

Step2: Analyze Vertical Stretch

The function \( g(x)=\frac{10}{x} \) can be written as \( g(x) = 10 \cdot \frac{1}{x} \), which is a vertical stretch of the parent function \( f(x)=\frac{1}{x} \) by a factor of 10. For a rational function \( y = \frac{k}{x} \) (where \( k>0 \)), as \( |x| \) increases, \( |y| \) decreases, and as \( |x| \) decreases, \( |y| \) increases. A larger \( |k| \) means the graph is "stretched" vertically, so the branches are farther from the x - axis and y - axis compared to \( y=\frac{1}{x} \) or \( y = \frac{5}{x} \) (for example). We need to identify the graph of \( y=\frac{10}{x} \) among the given options. Since we know the shape is a hyperbola in Q1 and Q3 (because \( k = 10>0 \)) and it's a vertical stretch of \( y=\frac{1}{x} \), we look for the hyperbola with branches that are more "spread out" (farther from the axes) compared to the parent function's typical graph.
(Note: Since the actual graphs are not fully visible, but based on the function's properties, we can determine the correct graph by its shape and the effect of the vertical stretch. If we assume the three graphs are of hyperbolas with different \( k \) values, the one with \( k = 10 \) will have the branches that are the most stretched vertically, i.e., for a given \( x \) (non - zero), the \( y \) - value is the largest among the three functions (if the other two have smaller \( k \) values like \( k = 1,k = 5 \) etc.))

Answer:

(Assuming the first graph is the one with the hyperbola of \( y=\frac{10}{x} \) based on vertical stretch. If we consider the three graphs from left to right, the left - most graph (or the one with the most vertically stretched hyperbola in Q1 and Q3) is the graph of \( g(x)=\frac{10}{x} \). So the answer would be "The Left Graph" (or the specific graph that shows the hyperbola of \( y = \frac{10}{x} \) with two branches in Q1 and Q3 and a vertical stretch of factor 10 from \( y=\frac{1}{x} \)).)