Question

match each function with its graph. (one choice will not be used.)
drag each graph above to match the function below.
$y = \sin(x - \frac{\pi}{3})$ $y = \sin(x + \frac{\pi}{3})$ $y = \cos(x - \frac{\pi}{3})$ $y = \cos(x + \frac{\pi}{3})$ $y = 1+\sin x$ $y=-1 + \sin x$ $y = 1+\cos x$ $y=-1 + \cos x$

Explanation:

Step1: Recall sine - cosine graph properties

The general form of a sinusoidal function is $y = A\sin(B(x - C))+D$ or $y = A\cos(B(x - C))+D$, where $A$ is the amplitude, $B$ affects the period ($T=\frac{2\pi}{|B|}$), $C$ is the phase - shift, and $D$ is the vertical shift.
For $y = \sin(x-\frac{\pi}{3})$, the graph of $y = \sin x$ is shifted to the right by $\frac{\pi}{3}$ units. The zero - crossing that was at $x = 0$ for $y=\sin x$ is now at $x=\frac{\pi}{3}$.
For $y=\sin(x + \frac{\pi}{3})$, the graph of $y = \sin x$ is shifted to the left by $\frac{\pi}{3}$ units. The zero - crossing that was at $x = 0$ for $y=\sin x$ is now at $x=-\frac{\pi}{3}$.
For $y=\cos(x-\frac{\pi}{3})$, the graph of $y = \cos x$ is shifted to the right by $\frac{\pi}{3}$ units. The maximum that was at $x = 0$ for $y=\cos x$ is now at $x=\frac{\pi}{3}$.
For $y=\cos(x+\frac{\pi}{3})$, the graph of $y = \cos x$ is shifted to the left by $\frac{\pi}{3}$ units. The maximum that was at $x = 0$ for $y=\cos x$ is now at $x=-\frac{\pi}{3}$.
For $y = 1+\sin x$, the graph of $y=\sin x$ is shifted up by 1 unit. The range is $[0,2]$.
For $y=-1+\sin x$, the graph of $y=\sin x$ is shifted down by 1 unit. The range is $[-2,0]$.
For $y = 1+\cos x$, the graph of $y=\cos x$ is shifted up by 1 unit. The range is $[0,2]$.
For $y=-1+\cos x$, the graph of $y=\cos x$ is shifted down by 1 unit. The range is $[-2,0]$.

Step2: Match the functions with the graphs

• $y=\sin(x - \frac{\pi}{3})$: The graph that has a zero - crossing at $x=\frac{\pi}{3}$ and has the shape of a sine function.
• $y=\sin(x+\frac{\pi}{3})$: The graph that has a zero - crossing at $x =-\frac{\pi}{3}$ and has the shape of a sine function.
• $y=\cos(x-\frac{\pi}{3})$: The graph that has a maximum at $x=\frac{\pi}{3}$ and has the shape of a cosine function.
• $y=\cos(x+\frac{\pi}{3})$: The graph that has a maximum at $x=-\frac{\pi}{3}$ and has the shape of a cosine function.
• $y = 1+\sin x$: The graph with a minimum of 0 and a maximum of 2 and the shape of a sine function.
• $y=-1+\sin x$: The graph with a minimum of - 2 and a maximum of 0 and the shape of a sine function.
• $y = 1+\cos x$: The graph with a minimum of 0 and a maximum of 2 and the shape of a cosine function.
• $y=-1+\cos x$: The graph with a minimum of - 2 and a maximum of 0 and the shape of a cosine function.

(Without the actual graph - to - function matching results shown in the question, we can't give the final specific match. But the above steps show how to approach the problem.)

Since we don't have the ability to directly see the specific graphs and match them, we can't provide a final answer in the form of "Graph 1 matches function A" etc. But the above steps outline the process of matching. If you can provide more details about the graphs (such as characteristics like intercepts, maxima, minima), we can complete the matching.

Answer:

Step1: Recall sine - cosine graph properties

The general form of a sinusoidal function is $y = A\sin(B(x - C))+D$ or $y = A\cos(B(x - C))+D$, where $A$ is the amplitude, $B$ affects the period ($T=\frac{2\pi}{|B|}$), $C$ is the phase - shift, and $D$ is the vertical shift.
For $y = \sin(x-\frac{\pi}{3})$, the graph of $y = \sin x$ is shifted to the right by $\frac{\pi}{3}$ units. The zero - crossing that was at $x = 0$ for $y=\sin x$ is now at $x=\frac{\pi}{3}$.
For $y=\sin(x + \frac{\pi}{3})$, the graph of $y = \sin x$ is shifted to the left by $\frac{\pi}{3}$ units. The zero - crossing that was at $x = 0$ for $y=\sin x$ is now at $x=-\frac{\pi}{3}$.
For $y=\cos(x-\frac{\pi}{3})$, the graph of $y = \cos x$ is shifted to the right by $\frac{\pi}{3}$ units. The maximum that was at $x = 0$ for $y=\cos x$ is now at $x=\frac{\pi}{3}$.
For $y=\cos(x+\frac{\pi}{3})$, the graph of $y = \cos x$ is shifted to the left by $\frac{\pi}{3}$ units. The maximum that was at $x = 0$ for $y=\cos x$ is now at $x=-\frac{\pi}{3}$.
For $y = 1+\sin x$, the graph of $y=\sin x$ is shifted up by 1 unit. The range is $[0,2]$.
For $y=-1+\sin x$, the graph of $y=\sin x$ is shifted down by 1 unit. The range is $[-2,0]$.
For $y = 1+\cos x$, the graph of $y=\cos x$ is shifted up by 1 unit. The range is $[0,2]$.
For $y=-1+\cos x$, the graph of $y=\cos x$ is shifted down by 1 unit. The range is $[-2,0]$.

Step2: Match the functions with the graphs

• $y=\sin(x - \frac{\pi}{3})$: The graph that has a zero - crossing at $x=\frac{\pi}{3}$ and has the shape of a sine function.
• $y=\sin(x+\frac{\pi}{3})$: The graph that has a zero - crossing at $x =-\frac{\pi}{3}$ and has the shape of a sine function.
• $y=\cos(x-\frac{\pi}{3})$: The graph that has a maximum at $x=\frac{\pi}{3}$ and has the shape of a cosine function.
• $y=\cos(x+\frac{\pi}{3})$: The graph that has a maximum at $x=-\frac{\pi}{3}$ and has the shape of a cosine function.
• $y = 1+\sin x$: The graph with a minimum of 0 and a maximum of 2 and the shape of a sine function.
• $y=-1+\sin x$: The graph with a minimum of - 2 and a maximum of 0 and the shape of a sine function.
• $y = 1+\cos x$: The graph with a minimum of 0 and a maximum of 2 and the shape of a cosine function.
• $y=-1+\cos x$: The graph with a minimum of - 2 and a maximum of 0 and the shape of a cosine function.

(Without the actual graph - to - function matching results shown in the question, we can't give the final specific match. But the above steps show how to approach the problem.)

Since we don't have the ability to directly see the specific graphs and match them, we can't provide a final answer in the form of "Graph 1 matches function A" etc. But the above steps outline the process of matching. If you can provide more details about the graphs (such as characteristics like intercepts, maxima, minima), we can complete the matching.