Question

graph the function by applying an appropriate reflection.
$k(x) = -\frac{1}{x}$

Explanation:

Step1: Identify parent function

The parent function is $f(x)=\frac{1}{x}$.

Step2: Recognize reflection type

The given function $k(x)=-\frac{1}{x}$ is a reflection of $f(x)$ over the $x$-axis (or $y$-axis, as both produce the same result for this function). For any point $(x,y)$ on $f(x)$, the corresponding point on $k(x)$ is $(x,-y)$.

Step3: Plot key reflected points

• For $f(1)=1$, $k(1)=-1$ → point $(1,-1)$
• For $f(2)=\frac{1}{2}$, $k(2)=-\frac{1}{2}$ → point $(2,-\frac{1}{2})$
• For $f(-1)=-1$, $k(-1)=1$ → point $(-1,1)$
• For $f(-2)=-\frac{1}{2}$, $k(-2)=\frac{1}{2}$ → point $(-2,\frac{1}{2})$

Step4: Draw the reflected hyperbola

Sketch the two branches of the hyperbola passing through the reflected points, with asymptotes at the $x$-axis ($y=0$) and $y$-axis ($x=0$).

Answer:

The graph of $k(x)=-\frac{1}{x}$ is the hyperbola of $y=\frac{1}{x}$ reflected over the $x$-axis (or $y$-axis), with branches in the second and fourth quadrants, asymptotes at $x=0$ and $y=0$, passing through points like $(-1,1)$, $(1,-1)$, $(-2,\frac{1}{2})$, $(2,-\frac{1}{2})$.