Question

without graphing, determine whether the equation y = x²+5 has a graph that is symmetric with respect to the x - axis, the y - axis, the origin, or none of these. choose the correct answer below. select all that apply. a. x - axis b. y - axis c. origin d. none of these

Explanation:

Step1: Test for x - axis symmetry

Replace \(y\) with \(-y\) in the equation \(y = x^{2}+5\). We get \(-y=x^{2}+5\) or \(y=-x^{2} - 5\), which is not the same as the original equation. So, it is not symmetric about the x - axis.

Step2: Test for y - axis symmetry

Replace \(x\) with \(-x\) in the equation \(y = x^{2}+5\). We have \(y=(-x)^{2}+5=x^{2}+5\), which is the same as the original equation. So, it is symmetric about the y - axis.

Step3: Test for origin symmetry

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(y = x^{2}+5\). We get \(-y=(-x)^{2}+5\), or \(y=-x^{2}-5\), which is not the same as the original equation. So, it is not symmetric about the origin.

Answer:

B. y - axis