13) [tex]y=(x-1)^3(x+2)^2[/tex] changes sign
x=1 bl x=2 c| x=-2
C) [tex]x=-1[/tex]
14) What is [tex]y[/tex]-intercept?
[tex]Y=-(x+1)^2(x-3)(x-2)[/tex]
a) [tex](0,6)[/tex]
di [tex](0,-5)[/tex]
b) [tex](0,5)[/tex]
(c) [tex](0,-6)[/tex]
Answer (2)
The function y = ( x − 1 ) 3 ( x + 2 ) 2 changes sign at x = 1 . The y-intercept of the function y = − ( x + 1 ) 2 ( x − 3 ) ( x − 2 ) is ( 0 , − 6 ) . Therefore, the answer for the y-intercept is option (c) (0,-6).
;
The function y = ( x − 1 ) 3 ( x + 2 ) 2 changes sign at x = 1 .
The y-intercept of y = − ( x + 1 ) 2 ( x − 3 ) ( x − 2 ) is found by setting x = 0 .
Substituting x = 0 gives y = − ( 1 ) 2 ( − 3 ) ( − 2 ) = − 6 .
The y-intercept is ( 0 , − 6 ) .
Explanation
Problem Analysis We are given two separate math problems. The first asks where the function y = ( x − 1 ) 3 ( x + 2 ) 2 changes sign, and the second asks for the y-intercept of the function y = − ( x + 1 ) 2 ( x − 3 ) ( x − 2 ) . We will address each problem individually.
Sign Change Analysis For the first problem, we need to determine where the function y = ( x − 1 ) 3 ( x + 2 ) 2 changes sign. A function changes sign at a root if the root has odd multiplicity. The factor ( x − 1 ) 3 has a root at x = 1 with multiplicity 3, which is odd. Therefore, the function changes sign at x = 1 . The factor ( x + 2 ) 2 has a root at x = − 2 with multiplicity 2, which is even. Therefore, the function does not change sign at x = − 2 . The option x = − 1 is not a root of the function, so it is not relevant to sign changes.
Conclusion for Question 13 Therefore, the function changes sign at x = 1 .
Y-Intercept Calculation For the second problem, we need to find the y-intercept of the function y = − ( x + 1 ) 2 ( x − 3 ) ( x − 2 ) . The y-intercept occurs when x = 0 . Substituting x = 0 into the equation, we get y = − ( 0 + 1 ) 2 ( 0 − 3 ) ( 0 − 2 ) = − ( 1 ) 2 ( − 3 ) ( − 2 ) = − ( 1 ) ( 6 ) = − 6 .
Conclusion for Question 14 Therefore, the y-intercept is ( 0 , − 6 ) .
Examples
Understanding where a function changes sign is crucial in many real-world applications, such as analyzing population growth models or determining the stability of physical systems. For example, in population modeling, the sign change of a function might indicate a transition from population growth to decline. Finding the y-intercept of a function is equally important, as it often represents an initial value or a starting point in a given scenario. For instance, in a savings account model, the y-intercept could represent the initial deposit amount.
