The graph of $$ y=x^{2} $$ opens up and the graph of $$ y=-x^{2} $$ opens down. How would you compare the graphs of the following pairs of equations? a) $$ y=\sqrt{x} $$ and $$ y=-\sqrt{x} $$ b) $$ y=|x| $$ and $$ y=-|x| $$ c) $$ y=\frac{1}{x} $$ and $$ y=-\frac{1}{x} $$

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**a) Comparing $$ y=\sqrt{x} $$ and $$ y=-\sqrt{x} $$:**

- The graph of $$ y=\sqrt{x} $$ has:
  - **Domain:** $$ x \ge 0 $$
  - **Range:** $$ y \ge 0 $$
  - It starts at the origin, $$ (0, 0) $$, and increases slowly as $$ x $$ increases.

- The graph of $$ y=-\sqrt{x} $$ is the reflection of $$ y=\sqrt{x} $$ in the $$ x $$-axis:
  - **Domain:** $$ x \ge 0 $$
  - **Range:** $$ y \le 0 $$
  - It starts at $$ (0, 0) $$ and decreases (goes downward) as $$ x $$ increases.

Thus, the two graphs are mirror images of each other with respect to the $$ x $$-axis.

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**b) Comparing $$ y=|x| $$ and $$ y=-|x| $$:**

- The graph of $$ y=|x| $$ forms a "V" shape opening upwards:
  - **Domain:** All real numbers.
  - **Range:** $$ y \ge 0 $$
  - Its vertex is at the origin, $$ (0, 0) $$.

- The graph of $$ y=-|x| $$ is obtained by reflecting $$ y=|x| $$ across the $$ x $$-axis:
  - **Domain:** All real numbers.
  - **Range:** $$ y \le 0 $$
  - Its vertex is also at $$ (0, 0) $$, but the "V" now opens downward.

Thus, these graphs are reflections of one another about the $$ x $$-axis.

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**c) Comparing $$ y=\frac{1}{x} $$ and $$ y=-\frac{1}{x} $$:**

- The graph of $$ y=\frac{1}{x} $$ is a hyperbola with:
  - **Domain:** All real numbers except $$ x=0 $$.
  - **Range:** All real numbers except $$ y=0 $$.
  - Its branches lie in the first quadrant (where both $$ x $$ and $$ y $$ are positive) and the third quadrant (where both $$ x $$ and $$ y $$ are negative).

- The graph of $$ y=-\frac{1}{x} $$ is the reflection of $$ y=\frac{1}{x} $$ with respect to the $$ x $$-axis:
  - **Domain:** All real numbers except $$ x=0 $$.
  - **Range:** All real numbers except $$ y=0 $$.
  - Its branches lie in the second quadrant (where $$ x $$ is negative and $$ y $$ is positive) and the fourth quadrant (where $$ x $$ is positive and $$ y $$ is negative).

Thus, while both are hyperbolas with similar asymptotes ($$ x=0 $$ and $$ y=0 $$), $$ y=-\frac{1}{x} $$ is obtained by reflecting $$ y=\frac{1}{x} $$ across the $$ x $$-axis.
**a) $$ y=\sqrt{x} $$ and $$ y=-\sqrt{x} $$:** These graphs are mirror images of each other across the x-axis. **b) $$ y=|x| $$ and $$ y=-|x| $$:** These graphs are mirror images of each other across the x-axis. **c) $$ y=\frac{1}{x} $$ and $$ y=-\frac{1}{x} $$:** These graphs are mirror images of each other across the x-axis.

The graph of opens up and the graph of opens down. How
would you compare the graphs of the following pairs of equations?
a) and
b) and
c) and

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