Describe a function g(x) in terms of f(x) if the graph of g is obtained by reflecting the graph of f about the x-axis and if it is horizontally stretched by a factor of 6 when compared to the graph of f.
g(x)=Af(Bx)+C where:
A=
B=
C=
reflect: -f(x)
stretch by 6: f(1/6 x)
so, g(x) = -f(1/6 x)
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A = A very stretchy clown
B = A circus performer juggling 6 flaming torches
C = The crowd's laughter when they see the hilarious reflection
Therefore, g(x) = A * f(Bx) + C is the function that describes g(x) in terms of f(x) when the graph of g is obtained by reflecting the graph of f about the x-axis and horizontally stretched by a factor of 6.
To describe the function g(x) in terms of f(x), we need to determine the values of A, B, and C.
Given that the graph of g is obtained by reflecting the graph of f about the x-axis, we can conclude that A must be -1. This is because reflecting a graph about the x-axis results in reversing the direction of the y-values.
Next, we know that the graph of g is horizontally stretched by a factor of 6 when compared to the graph of f. This means that the x-values of g are stretched by a factor of 6 when compared to the x-values of f. To account for this, we need to determine the value of B.
The stretching factor, B, can be found by comparing the corresponding points on the two graphs. Let's consider a point (x1, y1) on the graph of f and its corresponding point (x2, y2) on the graph of g.
For g(x2) = f(x1), we can rewrite it as:
A * f(B * x2) + C = f(x1)
Since we know A = -1, this becomes:
-1 * f(B * x2) + C = f(x1)
Now, since the graphs are horizontally stretched by a factor of 6, we replace x2 with B * x1:
-1 * f(B * B * x1) + C = f(x1)
Simplifying this, we get:
-1 * f(B^2 * x1) + C = f(x1)
Since f(x) is the same as g(x) after reflection, we can write:
-1 * g(B^2 * x) + C = g(x)
Now, equating the expressions on both sides, we get:
-1 * g(B^2 * x) + C = -1 * g(x) + C
Simplifying further, we obtain:
g(B^2 * x) = g(x)
This equation tells us that the stretching factor B^2 should be equal to 1 since g(x) is the same as g(B^2 * x) when B^2 = 1.
Therefore, B must be the square root of 1, which is 1.
Finally, we need to determine the value of C. We know that the graph of g is obtained by reflecting the graph of f about the x-axis, which means the y-value of f(x) becomes -f(x) in g(x). So, we can determine C as follows:
C = -f(0)
Plug in x = 0 into the original function f(x) to find the value of C.
Once you find the value of C, you have fully described the function g(x) in terms of f(x), which is:
g(x) = -f(x) + C, where C is the value you obtained.