Give example of each function:
(a) degree 2; two zeros, the function decreases then increases
(b) degree 3; three zeros, the function increases, decreases, then increases
(c) degree 3; two zeros, the function decreases, increases, then decreases
(d) degree 1; one zero, the function increases
To find examples of each function, we need to consider the degree of the function and the behavior of the function in terms of zeros and its variation (increase or decrease).
(a) Degree 2; two zeros, the function decreases then increases:
An example of this type of function is a quadratic function. Let's consider the function f(x) = x^2 - 4x + 3. To find the zeros, we set the function equal to zero and solve for x:
x^2 - 4x + 3 = 0
(x - 1)(x - 3) = 0
So, the zeros of the function are x = 1 and x = 3. If we graph this function, we will see that it starts by decreasing (from positive to negative) until x = 1, then it starts to increase (from negative to positive) until x = 3.
(b) Degree 3; three zeros, the function increases, decreases, then increases:
An example of this type of function is a cubic function. Let's consider the function g(x) = x^3 - 3x^2 + 2x. To find the zeros, we set the function equal to zero and solve for x:
x^3 - 3x^2 + 2x = 0
x(x - 1)(x - 2) = 0
So, the zeros of the function are x = 0, x = 1, and x = 2. If we graph this function, we will see that it starts by increasing (from negative to positive) until x = 0, then it decreases (from positive to negative) until x = 1, and finally, it increases (from negative to positive) until x = 2.
(c) Degree 3; two zeros, the function decreases, increases, then decreases:
An example of this type of function is again a cubic function. Let's consider the function h(x) = -x^3 + 4x^2 - 3x. To find the zeros, we set the function equal to zero and solve for x:
-x^3 + 4x^2 - 3x = 0
x(x - 1)(x - 3) = 0
So, the zeros of the function are x = 0, x = 1, and x = 3. If we graph this function, we will see that it starts by decreasing (from positive to negative) until x = 1, then it increases (from negative to positive) until x = 3, and finally, it decreases (from positive to negative) after x = 3.
(d) Degree 1; one zero, the function increases:
An example of this type of function is a linear function. Let's consider the function j(x) = 2x - 1. To find the zero, we set the function equal to zero and solve for x:
2x - 1 = 0
2x = 1
x = 1/2
So, the zero of the function is x = 1/2. If we graph this function, we will see that it increases (from negative to positive) throughout the entire domain.