Questions: Without graphing, determine whether the equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. y=x^2+6 Select all that apply. A. x-axis B. y-axis C. origin D. none of these

Without graphing, determine whether the equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these.

y=x^2+6

Select all that apply.
A. x-axis
B. y-axis
C. origin
D. none of these

Transcript text: Without graphing, determine whether the equation has a graph that is symmetric with respect to the $x$-axis, the $y$-axis, the origin, or none of these. \[ y=x^{2}+6 \] Select all that apply. A. $x$-axis B. $y$-axis C. origin D. none of these

Solution

Solution Steps

Step 1: Determine the polynomial expression

The polynomial expression is: $P(x) = 1 + 0x + 6x^2$

Step 2: Check for symmetry with respect to the y-axis

To check for y-axis symmetry, we evaluate $P(-x) = 1 (-x) + (6)(-x)^2$ and simplify to $P(-x) = 1 - 0x + 6x^2$. Since $P(-x) eq P(x)$, the graph is not symmetric with respect to the y-axis.

Step 3: Check for symmetry with respect to the origin

To check for origin symmetry, we evaluate $P(-x) = 1 (-x) + (6)(-x)^2$ and compare it to $-P(x) = -1 - 0x - 6x^2$. Since $P(-x) eq -P(x)$, the graph is not symmetric with respect to the origin.

Final Answer:

The graph of the given polynomial is not symmetric with respect to the y-axis. The graph of the given polynomial is not symmetric with respect to the origin.

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