1. The transformations for the equation below compared to the parent function y = x ^ 3/ y = - 1/2 * (x + 4) ^ 3 - 1 is Vertical Stretch, Horizontal Shift and Vertical Shift.
a. It will have exactly 3 possible complex roots.
b. There is exactly 1 possible negative root.
c. The possible rational roots are ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.
d. By applying synthetic division with the possible roots, we can determine the actual zeros (roots) of the function.
To describe the transformations for the equation y = -1/2 * (x + 4) ^ 3 - 12 compared to the parent function y = x ^ 3, we identify the changes in the equation. Using the Fundamental Theorem of Algebra, we determine the possible complex roots.
Using Descartes Rule, we find the possible positive and negative roots, and using the Rational Roots Theorem, we list the possible rational roots for the function. Finally, using synthetic division with the possible roots, we find the actual zeros (roots) of the function.
To describe the transformations for the equation y = -1/2 * (x + 4) ^ 3 - 12 compared to the parent function y = x ^ 3, we need to identify the changes in the equation. Here are the steps:
- Vertical Stretch: The coefficient -1/2 before the parent function affects the y-values, causing a vertical compression by a factor of 1/2.
- Horizontal Shift: The addition of -4 inside the parentheses shifts the graph 4 units to the left.
- Vertical Shift: The subtraction of 12 at the end of the equation shifts the graph 12 units downward.
These transformations change the shape, size, and position of the graph compared to the original function.
a. The Fundamental Theorem of Algebra states that a polynomial equation of degree n (highest exponent) has exactly n complex (including real and imaginary) roots. Since the given equation is a cubic equation (degree 3), it will have exactly 3 possible complex roots.
b. Descartes' Rule of Signs helps determine the possible positive and negative roots of a polynomial equation. Counting the sign changes in the given equation, there are 2 sign changes. This means that there are either 2 or 0 positive roots. To find the possible negative roots, we evaluate the function at -x. Plugging in -x gives f(-x) = -3x^3 - 7x^2 + 14x + 24. Counting the sign changes in this function, there is 1 sign change. Therefore, there is exactly 1 possible negative root.
c. The Rational Roots Theorem helps us find the possible rational roots by considering the factors of the constant term (24) divided by the factors of the leading coefficient (3). The possible rational roots are ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.
d. To find the possible zeros (roots) using the possible rational roots found in the previous question, we use synthetic division. We divide the given function f(x) = 3x^3 - 7x^2 - 14x + 24 by each possible rational root. If the remainder is zero, it means that the possible root is a zero (root) of the function. By applying synthetic division with the possible roots, we can determine the actual zeros (roots) of the function.