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 provide examples for each function, let's first understand what is meant by "degree" and "zeros" in this context.
Degree: The degree of a polynomial function refers to the highest exponent of the variable(s) present in the function. For example, a polynomial with the highest exponent of 2 would be a quadratic function (degree 2), whereas the highest exponent of 3 would indicate a cubic function (degree 3), and so on.
Zeros: Zeros of a function are also known as roots or x-intercepts. They indicate the values of the independent variable (usually denoted as x) for which the function equals zero. In simpler terms, zeros are the values of x that make the function cross or touch the x-axis.
Now, let's go through each example:
(a) Degree 2 with two zeros, the function decreases then increases:
An example of this is a quadratic function, such as f(x) = (x-1)(x-3). In this case, the function is a parabola that opens upwards. It has two zeros at x = 1 and x = 3. As you move from left to right along the graph, the function decreases until it reaches the first zero at x = 1. After that, it increases until it reaches the second zero at x = 3.
(b) Degree 3 with three zeros, the function increases, decreases, then increases:
An example of this is a cubic function, such as f(x) = (x+1)(x-2)(x-4). In this example, the function has three zeros at x = -1, x = 2, and x = 4. As you move from left to right along the graph, the function increases until it reaches the first zero at x = -1. After that, it decreases until it reaches the second zero at x = 2, and then it increases again until it reaches the third zero at x = 4.
(c) Degree 3 with two zeros, the function decreases, increases, then decreases:
An example of this is another cubic function, such as f(x) = -(x+1)(x-2)(x-3). In this example, the function has two zeros at x = -1 and x = 3. When you move from left to right along the graph, the function decreases until it reaches the first zero at x = -1. Then it increases until it reaches the second zero at x = 3, and finally, it decreases again beyond x = 3.
(d) Degree 1 with one zero, the function increases:
An example of this is a linear function, such as f(x) = 2x + 3. In this case, the function has one zero at x = -3/2. As you move from left to right along the graph, the function steadily increases without ever decreasing.
Note: These are just examples of functions that match the given descriptions. There are many other possible functions that could fit the given criteria.