Determine whether the function is symmetric with respect to the y-axis, symmetric with respect to the x-axis, symmetric with respect to the origin, or none of these.
y = -6x3 + 7x
it is an odd function, so it is symmetric about (0,0)
To determine whether a function is symmetric with respect to the y-axis, we replace x with -x in the equation and check if the resulting equation is equivalent to the original equation.
For the given function y = -6x^3 + 7x, if we replace x with -x, we get y = -6(-x)^3 + 7(-x).
Simplifying this, we have y = -6(-x^3) - 7x, which simplifies further to y = -6x^3 - 7x.
Comparing this with the original equation y = -6x^3 + 7x, we can see that the signs in front of the terms are different. This means that the function is not symmetric with respect to the y-axis.
To determine whether a function is symmetric with respect to the x-axis, we replace y with -y in the equation and check if the resulting equation is equivalent to the original equation.
For the given function y = -6x^3 + 7x, if we replace y with -y, we get -y = -6x^3 + 7x.
Multiplying both sides by -1, we have y = 6x^3 - 7x.
Comparing this with the original equation, y = -6x^3 + 7x, we can see that the signs for both the terms are different. This means that the function is not symmetric with respect to the x-axis.
To determine whether a function is symmetric with respect to the origin (0,0), we replace x with -x and y with -y, and check if the resulting equation is equivalent to the original equation.
For the given function y = -6x^3 + 7x, if we replace x with -x and y with -y, we get -y = -6(-x)^3 + 7(-x).
Simplifying this, we have -y = -6(-x^3) - 7x, which simplifies further to y = 6x^3 - 7x.
Comparing this with the original equation, y = -6x^3 + 7x, we can see that the signs for both the terms are different. This means that the function is not symmetric with respect to the origin.
Therefore, the given function y = -6x^3 + 7x is not symmetric with respect to the y-axis, the x-axis, or the origin.