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!
consider A(x) = Ax^3+Bx^2+cx + D
A(-3)=0=-27A+9B-3C + D
A(0)=-2=D that is nice
A(1)=0=A+B+C+D
A(2)=....
so you can solve for the four constants A,B,C,D and you have an equation that fits the points....if all the points are allowed in the domain.
If C(x)=A(2x+3) domain is all values of x , and range is all numbers.
To find the domain and range of C(x), we first need to understand how C(x) is defined.
Let's start with the domain. The domain of C(x) is the set of all possible values of x for which C(x) is defined. In this case, C(x) is defined as A(2x+3), which means that we need to consider the domain of A(x) and find the values of x that satisfy 2x+3.
From the given points of A(x), the x-values are -3, 0, 1, and 2. We can see that 2x+3 will vary as we substitute these x-values.
For -3: 2(-3) + 3 = -6 + 3 = -3
For 0: 2(0) + 3 = 0 + 3 = 3
For 1: 2(1) + 3 = 2 + 3 = 5
For 2: 2(2) + 3 = 4 + 3 = 7
So, the domain of C(x) is the set of all real numbers that make 2x+3 equal to one of the values above. In this case, it would be {-3, 0, 1, 2}.
Now, let's move on to the range. The range of C(x) is the set of all possible values of C(x) for the values in the domain. To determine the range of C(x), we need to evaluate C(x) for each value in the domain.
Using the points given in A(x), we find the corresponding y-values as follows:
For (-3,0): C(-3) = A(2(-3)+3) = A(-3) = 0
For (0,-2): C(0) = A(2(0)+3) = A(3) = -2
For (1,0): C(1) = A(2(1)+3) = A(5) = 0
For (2,3): C(2) = A(2(2)+3) = A(7) = 3
The range of C(x) is the set of all possible y-values obtained from evaluating C(x) for each value in the domain. In this case, it would be {0, -2, 3}.
Now, let's discuss how the graph of C(x) would look like. Since C(x) is defined as A(2x+3), we can transform the graph of A(x) to obtain the graph of C(x).
To obtain the graph of C(x), we need to shift the graph of A(x) horizontally by 3 units to the left (since we have 2x + 3). This means that all the x-values will be shifted by -3.
The graph of C(x) will have the same y-values as A(x), but the x-values will be shifted to the left by 3 units. The points on the graph of C(x) would be:
(-6,0), (-3,-2), (0,0), and (1,3), corresponding to the (-3,0), (0,-2), (1,0), and (2,3) points on A(x).
Regarding the equation of A(x) with the given points, there are a few different methods to find the equation. One approach is to use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.
To find the equation of A(x), we can use the two points (-3,0) and (2,3). We can calculate the slope as:
m = (y2 - y1)/(x2 - x1) = (3 - 0)/(2 - (-3)) = 3/5
Now, we can substitute the slope (m) and one of the points (let's use (-3,0)) in the slope-intercept form equation and solve for b:
0 = (3/5)(-3) + b
0 = -9/5 + b
b = 9/5
The equation of A(x) would be:
A(x) = (3/5)x + 9/5
Please note that this is just one possible equation for the given points, and there may be other equations that also satisfy the same set of points.