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 $$ y=-\frac{1}{x} $$

Question

UpStudy AI · Accepted Answer

**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.

---

**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.

---

**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

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Pre Calculus Questions

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