Answer:
An x-intercept is where y is zero. With the law of mutiplying by zero, we know that:
If a * b = 0, then a, b, or both must be zero.
So if we want that a function will be zero at 4 and 2, we need to find a function which equals zero at these points.
x = 2 / subtract 2 from both sides.
x - 2 = 0
So the expression is (x - 2).
The expression for x = 4 is (x - 4).
Now we just multiply them and give the function a name:
y = (x - 2)(x - 4)
y = x^2 - 6x + 8.
2. Same idea:
y = (x + 6)(x - 3)
y = x^2 +3x - 18.
3. Once again, same idea:
y = (x + 5)(x - 5)(x - 1)
Since (x + 5)(x - 5) = x^2 - 25, we can calculate:
y = (x^2 -25)(x - 1)
y = x^3 - x^2 -25x + 25.
4. As explained above, yes.
5. Yes, there is an infinite number of functions that fit the conditions of those questions.
Just multiply the same factor over and over:
For 1:
y = (x - 2)(x - 2)(x - 4)
y = (x^2 - 6x + 8)(x - 2)
y = x^3 - 8x^2 + 20x - 16.
The same goes for 2 and 3.
