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
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Solution

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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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