I don't understand my problems that I am suppose to do. please help.
Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or -1 and make the degree of the function as small as possible.
1.Crosses the x-axis at -4, 0, and 1; lies above the x-axis between -4 and 0; lies below the x-axis between 0 and 1.
3. Touches the x-axis at 0 and crosses the x-axis at 4; lies below the x-axis between 0 and 4
To write the equation of a polynomial function, we need to consider the given characteristics and determine the factors that contribute to them.
For the first problem, we know that the function crosses the x-axis at -4, 0, and 1. This means that the factors of the polynomial are (x + 4), x, and (x - 1). Additionally, it lies above the x-axis between -4 and 0, and below the x-axis between 0 and 1. This indicates that the factor (x - 1) contributes to the function being negative between 0 and 1.
To find the equation of the polynomial, we multiply these factors together:
f(x) = (x + 4)(x)(x - 1)
Expanding this expression, we get:
f(x) = (x^2 + 4x)(x - 1)
Now, simplifying it further:
f(x) = x^3 - x^2 + 4x^2 - 4x
Combining like terms, we get the final equation:
f(x) = x^3 + 3x^2 - 4x
For the second problem, we are told that the polynomial touches the x-axis at 0 and crosses it at 4. This means that the factorization of the polynomial is x(x - 4). Additionally, it lies below the x-axis between 0 and 4, indicating that the factor (x - 4) contributes to the function being negative between 0 and 4.
The equation of the polynomial is then:
f(x) = x(x - 4)
Expanding this expression, we get:
f(x) = x^2 - 4x
Thus, the equation of the polynomial is f(x) = x^2 - 4x.
Remember to always consider the given characteristics and analyze how they relate to the factors of the polynomial when determining the equation.