for the given equation list the intercepts and test for symmetry
y = x^3 -512
You know what the graph of x^3 looks like, so this one is just the same curve shifted down by 512 units.
One x-intercept at x=8
one y-intercept at y=-512
no symmetry, since the function is neither even nor odd.
Actually, there is radial symmetry about the point (0,-512) but that is not usually the intent.
To find the intercepts of the equation y = x^3 - 512, we need to set the value of y to zero and solve for x.
1. x-intercepts: To find the x-intercepts, we set y = 0 and solve for x. So, we have:
0 = x^3 - 512
To solve this equation, we can factor it by recognizing that 512 is equal to 8^3:
0 = (x - 8)(x^2 + 8x + 64)
Setting each factor equal to zero, we find:
x - 8 = 0 or x^2 + 8x + 64 = 0
Solving the first equation, we get:
x = 8
For the second equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values a = 1, b = 8, and c = 64, we have:
x = (-8 ± √(8^2 - 4(1)(64))) / (2(1))
x = (-8 ± √(64 - 256)) / (2)
x = (-8 ± √(-192)) / (2)
x = (-8 ± √(192)i) / (2)
x = (-8 ± 8√3 i) / (2)
x = -4 ± 4√3 i
Therefore, the x-intercepts of the equation are x = 8 and x = -4 ± 4√3 i.
2. y-intercept: To find the y-intercept, we set x = 0 and solve for y. So, we have:
y = (0)^3 - 512
y = -512
Therefore, the y-intercept of the equation is y = -512.
Now, let's test for symmetry:
3. Symmetry:
- Even symmetry: A function is even if it is symmetric about the y-axis. We test for even symmetry by substituting -x for x in the equation and checking if it remains the same.
Let's substitute -x for x in y = x^3 - 512:
y = (-x)^3 - 512
y = -x^3 - 512
Since y = -x^3 - 512 is not equivalent to y = x^3 - 512, the equation does not exhibit even symmetry.
- Odd symmetry: A function is odd if it is symmetric about the origin. We test for odd symmetry by substituting -x for x in the equation and checking if it becomes the negative of the original equation.
Let's substitute -x for x in y = x^3 - 512:
y = (-x)^3 - 512
y = -x^3 - 512
Since y = -x^3 - 512 is the negative of y = x^3 - 512, the equation exhibits odd symmetry.
To summarize:
- The x-intercepts of the equation y = x^3 - 512 are x = 8 and x = -4 ± 4√3 i.
- The y-intercept of the equation is y = -512.
- The equation exhibits odd symmetry.