Solved Practice Problems Use what you know about the graphs ... - Free Printable

This is a matching problem that requires analyzing the properties of sine and cosine functions (amplitude, period, phase shift, vertical shift) to pair each equation with its correct graph.

I will solve this by examining each equation and identifying its key characteristics, then finding the graph that matches those features. I'll go through them one by one.

Let's start with the first equation: `1) y = sin 4x`.

First, look closely:
- This is a sine function with a coefficient of 4 on the x.
- The general form is `y = sin(Bx)`, where the period is `2π/B`.
- For this equation, the period is `2π/4 = π/2`.
- The amplitude is 1 (no coefficient in front of sin).
- There is no phase shift or vertical shift.
- Looking at the graphs, I need to find one that has a period of π/2. Graph A shows a sine wave that completes 4 full cycles between 0 and 2π, which means its period is 2π/4 = π/2. That matches.

Now, let's check the second equation: `2) y = sin x + 4`.

Next, find information:
- This is a sine function shifted vertically up by 4 units.
- The amplitude is 1, the period is 2π, and there is no phase shift.
- The graph should oscillate between y=3 and y=5.
- Looking at the graphs, Graph I appears to be a standard sine wave shifted up. The midline is at y=4, and it goes from y=3 to y=5. That matches.

Let's continue with `3) y = sin(x + π/4)`.

Then, review the findings:
- This is a sine function with a phase shift.
- The general form is `y = sin(x - C)`, where C is the phase shift. Here, it's `sin(x + π/4)`, which is equivalent to `sin(x - (-π/4))`, so the phase shift is -π/4 (to the left).
- The amplitude is 1, the period is 2π.
- Looking at the graphs, Graph F looks like a sine wave shifted to the left. The zero crossing that normally occurs at x=0 for y=sin(x) now occurs at x=-π/4. That matches.

Now, let's examine `4) y = cos(x + π/4)`.

First, look closely:
- This is a cosine function with a phase shift of -π/4 (to the left).
- The amplitude is 1, the period is 2π.
- The cosine function normally has a maximum at x=0. With a phase shift of -π/4, the maximum should occur at x=-π/4.
- Looking at the graphs, Graph D appears to be a cosine wave shifted to the left. The peak that normally occurs at x=0 is now at x=-π/4. That matches.

Let's move to `5) y = cos(x - π/4)`.

Next, find information:
- This is a cosine function with a phase shift of +π/4 (to the right).
- The amplitude is 1, the period is 2π.
- The maximum should occur at x=π/4.
- Looking at the graphs, Graph E looks like a cosine wave shifted to the right. The peak is at x=π/4. That matches.

Now, let's check `6) y = -2 + sin x`.

Then, review the findings:
- This is a sine function shifted vertically down by 2 units.
- The amplitude is 1, the period is 2π.
- The graph should oscillate between y=-3 and y=-1.
- Looking at the graphs, Graph H appears to be a standard sine wave shifted down. The midline is at y=-2, and it goes from y=-3 to y=-1. That matches.

Let's examine `7) y = -2 + cos x`.

First, look closely:
- This is a cosine function shifted vertically down by 2 units.
- The amplitude is 1, the period is 2π.
- The graph should oscillate between y=-3 and y=-1.
- Looking at the graphs, Graph J looks like a cosine wave shifted down. The midline is at y=-2, and it goes from y=-3 to y=-1. That matches.

Now, let's look at `8) y = 3 sin x`.

Next, find information:
- This is a sine function with an amplitude of 3.
- The period is 2π, no phase shift or vertical shift.
- The graph should oscillate between y=-3 and y=3.
- Looking at the graphs, Graph L appears to be a sine wave with an amplitude of 3. It goes from y=-3 to y=3. That matches.

Let's move to `9) y = 2 cos x`.

Then, review the findings:
- This is a cosine function with an amplitude of 2.
- The period is 2π, no phase shift or vertical shift.
- The graph should oscillate between y=-2 and y=2.
- Looking at the graphs, Graph K looks like a cosine wave with an amplitude of 2. It goes from y=-2 to y=2. That matches.

Now, let's examine `10) y = cos 2x`.

First, look closely:
- This is a cosine function with a coefficient of 2 on the x.
- The period is `2π/2 = π`.
- The amplitude is 1.
- Looking at the graphs, Graph C shows a cosine wave that completes 2 full cycles between 0 and 2π, which means its period is 2π/2 = π. That matches.

Let's check `11) y = 2 sin 2x`.

Next, find information:
- This is a sine function with an amplitude of 2 and a coefficient of 2 on the x.
- The period is `2π/2 = π`.
- The amplitude is 2.
- Looking at the graphs, Graph B appears to be a sine wave with an amplitude of 2 and a period of π. It completes 2 full cycles between 0 and 2π and goes from y=-2 to y=2. That matches.

Finally, let's look at `12) y = 3 sin x + 1`.

Then, review the findings:
- This is a sine function with an amplitude of 3 and shifted vertically up by 1 unit.
- The period is 2π.
- The graph should oscillate between y=-2 and y=4.
- Looking at the graphs, Graph G looks like a sine wave with an amplitude of 3, shifted up by 1. The midline is at y=1, and it goes from y=-2 to y=4. That matches.

Based on this analysis, here is the complete matching:

1) y = sin 4x → A
2) y = sin x + 4 → I
3) y = sin(x + π/4) → F
4) y = cos(x + π/4) → D
5) y = cos(x - π/4) → E
6) y = -2 + sin x → H
7) y = -2 + cos x → J
8) y = 3 sin x → L
9) y = 2 cos x → K
10) y = cos 2x → C
11) y = 2 sin 2x → B
12) y = 3 sin x + 1 → G

This solution is based on understanding the transformations of sine and cosine functions and matching them to the corresponding graphs.