Question

Determine whether the graph of the equation is symmetric with respect to the​ y-axis, the​ x-axis, the​ origin, more than one of​ these, or none of these.
y equals 2 x plus 5
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Part 1
Select all that apply.
​x-axis
​y-axis
origin
none of these

Answer 1

To determine the symmetry of the graph of the equation \( y = 2x + 5 \), we can test for symmetry with respect to the y-axis, x-axis, and the origin.

1. **Y-axis symmetry**: A graph is symmetric with respect to the y-axis if replacing \( x \) with \( -x \) gives the original equation.
\[
y = 2(-x) + 5 = -2x + 5
\]
Since \( -2x + 5 \) is not the same as \( 2x + 5 \), the graph is not symmetric with respect to the y-axis.

2. **X-axis symmetry**: A graph is symmetric with respect to the x-axis if replacing \( y \) with \( -y \) gives the original equation.
\[
-y = 2x + 5 \quad \Rightarrow \quad y = -2x - 5
\]
Since \( -2x - 5 \) is not the same as \( 2x + 5 \), the graph is not symmetric with respect to the x-axis.

3. **Origin symmetry**: A graph is symmetric with respect to the origin if replacing \( x \) with \( -x \) and \( y \) with \( -y \) gives the original equation.
\[
-y = 2(-x) + 5 \quad \Rightarrow \quad -y = -2x + 5 \quad \Rightarrow \quad y = 2x - 5
\]
Since \( 2x - 5 \) is not the same as \( 2x + 5 \), the graph is not symmetric with respect to the origin.

Based on the tests:

- The graph is not symmetric with respect to the y-axis.
- The graph is not symmetric with respect to the x-axis.
- The graph is not symmetric with respect to the origin.

Thus, the correct selection is **none of these**.