y=x^2-8x

y=x(x-8)

x=0, 8

The equation provided is y = x^2 - 8x. This is a quadratic equation, and it represents a parabola when graphed. To understand the equation better, let's break it down and explain its components.

In a quadratic equation, the highest power of the variable is 2, represented by the term x^2. This term determines the shape of the parabola. The coefficient of x^2 is 1, which means the parabola opens upward because it is positive.

The next term is -8x, which represents the linear term. The coefficient of x is -8, indicating that the parabola will shift to the right or left. A negative coefficient means it will shift right, and a positive coefficient would mean a shift to the left.

To analyze the parabola in more detail, we can find the vertex, y-intercept, and x-intercepts.

1. Vertex:
To find the vertex of the parabola, we use the formula x = -b/2a. In this equation, a represents the coefficient of x^2, and b represents the coefficient of x. For our equation, a = 1 and b = -8.

x = -(-8) / 2(1)
x = 8/2
x = 4

Now, substitute this x-value back into the original equation to find the y-coordinate of the vertex.

y = (4)^2 - 8(4)
y = 16 - 32
y = -16

So, the vertex of the parabola is at (4, -16).

2. Y-intercept:
To find the y-intercept, substitute x = 0 into the equation.

y = (0)^2 - 8(0)
y = 0

Therefore, the y-intercept is 0, and the point is (0, 0).

3. X-intercepts:
To find the x-intercepts (also called zeros or roots), we set y = 0 and solve for x.

0 = x^2 - 8x

Using factoring or the quadratic formula, we find that the x-intercepts are x = 0 and x = 8.

Now, armed with this information, you can graph the equation y = x^2 - 8x and analyze its behavior.