The graph A(x) has points (-3,0), (0,-2), (1,0) and (2,3).
If C(x) is defined by C(x)=A(2x+3), then what is the domain and range of C and how would the graph of C(x) look like?
Also, is there any way to find what the actual equation of A(x) is with the points listed above? Thank you for the help!
To find the domain and range of C(x), let's first understand how the transformation C(x) = A(2x+3) affects the original function A(x).
1. DOMAIN OF C(x):
Since the transformation involves the expression 2x+3, it implies that x can take any real number. Therefore, the domain of C(x) is all real numbers.
2. RANGE OF C(x):
To find the range, we need to determine the range of A(x) and then apply the transformation. Looking at the given points of A(x) (-3,0), (0,-2), (1,0), and (2,3), we can see that the highest y-value is 3 and the lowest is -2. So, the range of A(x) is -2 ≤ y ≤ 3.
When we apply the transformation C(x) = A(2x+3), the x-values of A(x) are multiplied by 2 and shifted 3 units to the left. This means the range of C(x) will be the same as A(x) but shifted. Therefore, the range of C(x) is -2 ≤ y ≤ 3.
3. GRAPH OF C(x):
To graph C(x), we can start by graphing the original function A(x) using the given points. Joining those points will give a general idea of the shape of A(x) but not the exact equation without more information.
Regarding C(x), the transformation C(x) = A(2x+3) indicates that every x-value of C(x) corresponds to a specific x-value in A(x). By substituting (2x+3) into the equation of A(x), we can calculate the y-values of C(x). Using the points (-3,0), (0,-2), (1,0), and (2,3) of A(x), we can calculate the corresponding points of C(x).
For example, when x=-3:
C(x) = A(2(-3)+3) = A(-3)
Using the original points of A(x), we find that A(-3) = 0.
So, when x=-3, C(x) = 0.
Similar calculations can be done for other x-values. Once we have the corresponding points for C(x), we can join them to graph C(x).
Unfortunately, without more information, we cannot determine the exact equation of A(x) that matches the given points (-3,0), (0,-2), (1,0), and (2,3). These points can fit many different equations, and we would need additional information to narrow it down.