How does the graph of g(x) = cos(x) - 12 compare with the parent function g(x) = cos x

Answer 1

To understand how the graph of $g(x)=\cos (x)-12$ compares with the parent function $g(x)=\cos (x)$, we can analyze the transformations that have been applied to the parent function.

1. Parent Function: The parent function $g(x)=\cos (x)$ has a range of $[-1,1]$ and oscillates between these values with a period of $2\pi$.

2. Transformation: The function $g(x)=\cos (x)-12$ involves a vertical shift:

   • The term $-12$ indicates that the entire graph of $\cos (x)$ is shifted downward by 12 units.

3. New Range: After shifting down by 12, the new range of $g(x)=\cos (x)-12$ becomes:

   • The maximum value shifts from 1 to $1-12=-11$.
   • The minimum value shifts from -1 to $-1-12=-13$.
   • Therefore, the new range is $[-13,-11]$.

4. Period: The period of the cosine function is unchanged, so the period of $g(x)=\cos (x)-12$ remains $2\pi$.

5. Graph Comparison:

   • When comparing the two graphs, the graph of $g(x)=\cos (x)$ oscillates between -1 and 1 centered around the x-axis.
   • The graph of $g(x)=\cos (x)-12$ will oscillate between -13 and -11, centered around the line $y=-12$.

Conclusion: The graph of $g(x)=\cos (x)-12$ is the same shape as the graph of the parent function $g(x)=\cos (x)$, with both graphs oscillating with a period of $2\pi$, but the entire graph is moved down vertically by 12 units.