please help! functions

Question

Graph the transformations that are applied to 𝑦 = sin 𝑥 to obtain the sinusoidal function:

𝑓(𝑥) = 2 sin(𝑥 + 30°) + 2

Be sure to use the “key points” to graph your base function. GRAPH TWO FULL CYCLES

1. Label each stage of the transformations clearly
2. List the transformations applied to 𝑦 = sin 𝑥 to obtain the sinusoidal function assigned to you
3. State the domain & range
4. State the amplitude, equation of the axis, and max & min of your function
5. Label your scale accurately
6. Use a ruler when drawing grid lines (if necessary)
7. Provide full, clear solutions, with properly identified answers

Answer 1

Hi Shanell!

There are going to be 3 stages of transformations to get from  𝑦 = sin 𝑥 to 𝑓(𝑥) = 2 sin(𝑥 + 30°) + 2.

1. 
   
2. Multiply by 2: y = 2 * (sin(x))
3. this makes the sine function twice as high and twice as deep (what would this then make the amplitude, if the amplitude is 1 for sin(x)?)
4. Shift the phase to the left by 30 degrees: y = 2sin(x + 30°)
5. this "moves" the function to the left. A good way to think about this is: when x is 0, what is sin(x)? Now when x is 30 degrees what is sin(x)? Thus sin(x+30) is the same as moving the right along the x axis and increasing the value of sin(x) by 30 degrees.
6. Vertical shift of 3 units: y = (2 sin(𝑥 + 30°)) + 2
7. this "moves" the function up two units. This uses the same logic as 1.b, since sin(0) is normally 0, now sin(0)+2 = 2.
8. 
   
9. Multiply by 2: y = 2 * (sin(x))
10. Shift the phase to the left by 30 degrees: y = 2sin(x + 30°)
11. Vertical shift of 3 units: y = (2 sin(𝑥 + 30°)) + 2
12. 
    
13. Domain: These are the x's we can input into the function and get a valid output. The sine function is defined for all real numbers, and adding, subtracting, or multiplying real numbers does not change this.
14. The range of the function refers to all possible output values. For the given function, the range is the set of all real numbers from -2 to 4. The reason is that the sine function always produces outputs between -1 and 1. When we multiply by 2, the outputs are between -2 and 2, and when we then add 2, the outputs are between 0 and 4. Therefore, the range of the function is [0, 4].
15. a
16. Amplitude: The amplitude of a function is half the distance from between the max and the min. The normal amplitude of sin(x) is 1 since sin has a min of -1 and a max of 1. The distance between 1 and -1 is 2, and half of this equals 1. Hint: Now that our min is 0 and our max is 4, the amplitude will be half the distance from 0 to 4
17. Equation of the axis: Also known as the axis of symmetry. When you look at the graph of the function, where can you draw a single line that splits the function symmetrically in half (in other words, what line is halfway between the maximum of 4 and the minimum of 0)?
18. Max and min have been discussed in 3b and 4a,4b

When graphing the function, the "key points" you want to look out for are the maximums, minimums, zero-crossings, and period completion points (i.e. when the function makes a full cycle)

For the general sin(x) function they look like the following:

1. Max: x = -270degrees, 90degrees, sin(x) = 1
2. Min x = -90degrees, 270degrees, sin(x) = -1
3. Zero Crossings (x intercepts): x = +-180degrees, sin(x) = 0
4. Period Completion points: x = +-360, sin(x) = 0

Now we will need to apply our transformations to these in order to see what they should be for 𝑓(𝑥) = 2 sin(𝑥 + 30°) + 2

1. Max: when x = -270degrees and 90 degrees multiply (sin(x)) by 2 then add 2. Now shift these points left by 30 degrees (-300 degrees and 60 degrees)
2. Min: when x = -90degrees, 270degrees multiply sin(x) by 2 then add 2. Now shift these points left by 30 degrees.
3. Zero Crossings: since sin(x) = 0 when x = +-180 degrees, when does sin(x+30) = 0? Shift the points left by 30 degrees :)
4. Period Completion points: x = +-360, sin(x) = 0, we can now shift these points left by 30 degrees