Your parabola is described by this equation: f(x) = (x-2) (x-6) What relationship do you see between this function and its roots?
Perhaps that the roots, which are x = 2 and x = 6, happen to be where the parabola cuts the x-axis ?
Or it could be that the vertex would lie on a vertical line half way between
2 and 6 ?
Well, it seems like this parabola is like a very picky eater. It doesn't want any of its x-values to be either 2 or 6. It's like those roots are a big "nope" for this parabola. So, you could say that the relationship between this function and its roots is that they are like oil and water – they just don't mix!
The given equation represents a parabola described by the function f(x) = (x-2)(x-6). The roots, or x-intercepts, of a quadratic function are the values of x for which f(x) equals zero.
To find the roots of this function, we set f(x) equal to zero and solve for x:
0 = (x-2)(x-6)
To have a product equal to zero, at least one of the terms in the parentheses must be zero. So, we can set each factor equal to zero:
x-2 = 0 and x-6 = 0
Simplifying these equations, we find:
x = 2 and x = 6
Therefore, the roots of the function are x = 2 and x = 6.
The relationship between the function and its roots is that the function equals zero at the points where x = 2 and x = 6. These roots correspond to the x-intercepts of the parabola.
The equation of the parabola you provided is f(x) = (x - 2)(x - 6). To understand the relationship between the function and its roots, we need to consider what the roots represent.
The roots of a function are the values of x for which the function equals zero. In the case of a quadratic function, like the one given, the roots are also called the x-intercepts or zeros of the function. These are the points on the x-axis where the parabola intersects or touches the x-axis.
To find the roots of a quadratic function, we set the function equal to zero and solve for x. In this case, we set f(x) = 0:
0 = (x - 2)(x - 6)
Now, we can solve for x by using the zero product property. According to the property, if the product of two factors is equal to zero, then at least one of the factors must be zero. So we can set each factor equal to zero and solve for x:
x - 2 = 0 or x - 6 = 0
Adding 2 to both sides of the first equation, we get:
x = 2
Adding 6 to both sides of the second equation, we get:
x = 6
Therefore, the roots or x-intercepts of the given quadratic function are x = 2 and x = 6.
Now, let's analyze the relationship between the function and its roots. Since the given function is factorized into (x - 2)(x - 6), we can see that the roots 2 and 6 are the values that make each factor equal to zero.
When x = 2, the factor (x - 2) becomes zero, causing the entire function to equal zero. Similarly, when x = 6, the factor (x - 6) becomes zero, again resulting in the function equaling zero.
So, the relationship is that the roots (x-intercepts) of the function correspond to the values that make each factor of the function equal to zero.