Final answer:
To find the function and graph of the given polynomial with zeros and a y-intercept, we can write the function in factored form using the provided zeros and then simplify it to standard form. We can plot the zeros and the y-intercept on a graph and describe the end behavior using arrow notation. There are no zeros with multiplicities greater than one in the function.
Step-by-step explanation:
To find the function and the graph of the polynomial with the given zeros (-3, -4, 3, and 1) and y-intercept (0,4), we can start by writing the function in factored form. Since we have the zeros, we can write the equation as f(x) = a(x + 3)(x + 4)(x - 3)(x - 1), where a is the leading coefficient. To find the value of a, we can substitute the y-intercept into the equation. Since the y-intercept is (0,4), we have f(0) = a(0 + 3)(0 + 4)(0 - 3)(0 - 1) = 4. Solving this equation, we find that a = 4/36 = 1/9.
So, the function in factored form is f(x) = (1/9)(x + 3)(x + 4)(x - 3)(x - 1). To write the function in standard form, we multiply the factors and simplify:
f(x) = (1/9)(x^2 + 7x + 12)(x^2 - 4x - 3).
Next, we can graph the function by plotting the zeros and the y-intercept on a coordinate plane. The x-intercepts are -3, -4, 3, and 1, and the y-intercept is (0,4). Label the axes and all the intercepts on the graph.
The end behavior of the graph can be described using arrow notation. As x approaches negative infinity, the graph goes downward. As x approaches positive infinity, the graph goes upward.
Finally, we can determine if there are any zeros with multiplicities greater than one in the function. To do this, we can look for repeated factors in the factored form. In this case, there are no repeated factors, so there are no zeros with multiplicities greater than one.