Answer:
To determine if the given equations are symmetric with respect to the origin, we need to check if replacing x with -x and y with -y results in the same equation.
For the equations given:
y=7x-4 is not symmetric with respect to the origin since replacing x with -x and y with -y gives y=-7x+4, which is not the same equation.
y= -x+7 is not symmetric with respect to the origin since replacing x with -x and y with -y gives y=x-7, which is not the same equation.
y = -7x^2 is symmetric with respect to the origin since replacing x with -x and y with -y gives -y = -7(-x)^2, which simplifies to -y = -7x^2. Thus, the equation remains the same.
y = 6x^2 - 9 is not symmetric with respect to the origin since replacing x with -x and y with -y gives -y = 6(-x)^2 - 9, which simplifies to -y = 6x^2 - 9. Thus, the equation is not the same.
x=1/4 y^2 is not symmetric with respect to the origin since replacing x with -x and y with -y gives -x = 1/4 (-y)^2, which simplifies to -x = 1/4 y^2. Thus, the equation is not the same.
x = -y^2 + 9 is symmetric with respect to the origin since replacing x with -x and y with -y gives -x = -(-y)^2 + 9, which simplifies to -x = -y^2 + 9. Thus, the equation remains the same.
y=-1/6 x^3 is symmetric with respect to the origin since replacing x with -x and y with -y gives -y=-1/6(-x)^3, which simplifies to -y=-1/6 x^3. Thus, the equation remains the same.
y=x^3-1 is not symmetric with respect to the origin since replacing x with -x and y with -y gives -y=(-x)^3-1, which simplifies to -y=-x^3-1. Thus, the equation is not the same.
y=sqrt(x) is not symmetric with respect to the origin since replacing x with -x and y with -y gives -y=sqrt(-x), which is not the same equation.
y=sqrt(x)-6 is not symmetric with respect to the origin since replacing x with -x and y with -y gives -y=sqrt(-x)-6, which is not the same equation.
Therefore, the only equation that is symmetric with respect to the origin is y = -7x^2.