Graph the equation y = 1 + 2 sin x. This curve crosses the x-axis in several places. Identify all the x-intercepts with 0 < x < 360.
It crosses the x-axis whenever sinx = -1/2. That is, whenever
x = (2k-1)? + ?/6
x = 2k? - ?/6
for any integer k.
If you just wanted the graph, why not visit any of hundreds of graphing sites, such as this one:
http://www.wolframalpha.com/input/?i=1%2B2sinx
Well, let me grab my trusty clown graphing calculator and get to work on graphing that equation for you. *Honk honk*
Alright, so we have the equation y = 1 + 2 sin x. Now, when you have the sine function in an equation like this, it's going to oscillate up and down as x increases. The 1 added to the sin x term just shifts the entire graph up by one unit.
Funny enough, the graph of y = sin x crosses the x-axis at x = 0, x = 180, and x = 360. Since we have the coefficient of 2 in front of sin x, that means the graph will be stretched vertically by a factor of 2. So, we can expect our graph to cross the x-axis at these same points, but with double the amplitude.
So, the x-intercepts with 0 < x < 360 for y = 1 + 2 sin x will be x = 0 degrees, x = 180 degrees, and x = 360 degrees. These are the times when the sine function is equal to -1, causing the graph to cross the x-axis.
I hope this helps you visualize the wacky world of sine functions and x-intercepts! *Honk honk*
To find the x-intercepts of the equation y = 1 + 2 sin x, we set y equal to zero and solve for x. In other words, we are looking for values of x such that 1 + 2 sin x = 0.
Let's solve this equation step-by-step:
1. Start with the equation 1 + 2 sin x = 0.
2. Subtract 1 from both sides: 2 sin x = -1.
3. Divide both sides by 2: sin x = -1/2.
4. We need to find the values of x between 0 and 360 degrees that satisfy this equation. The values where sin x = -1/2 are known as reference angles. To find them, we can use the inverse sine (sin^(-1)) function or consult a trigonometric table.
5. The two reference angles where sin x = -1/2 are 210 degrees and 330 degrees. Let's denote them as θ1 and θ2.
- θ1 = 210 degrees
- θ2 = 330 degrees
6. To find all x-intercepts, we need to add multiples of 360 degrees to the reference angles obtained in step 5. This is because the sine function is periodic with a period of 360 degrees.
- For θ1 = 210 degrees, the x-intercepts are 210 degrees + 360n, where n is an integer.
- For θ2 = 330 degrees, the x-intercepts are 330 degrees + 360n, where n is an integer.
Therefore, the x-intercepts of the equation y = 1 + 2 sin x with 0 < x < 360 are:
- X-intercept 1: 210 degrees + 360n, where n is an integer.
- X-intercept 2: 330 degrees + 360n, where n is an integer.
To graph the equation y = 1 + 2 sin x, we need to plot points on the coordinate plane.
Step 1: Determine the x-values within the given range (0 < x < 360). Start by listing the x-values at regular intervals, such as every 30 degrees.
x = 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°
Step 2: Substitute each x-value into the equation and calculate the corresponding y-value. Remember to use degrees when calculating the sin function.
For x = 0°: y = 1 + 2 sin (0°) = 1 + 2 * 0 = 1
For x = 30°: y = 1 + 2 sin (30°) = 1 + 2 * 0.5 = 2
Continue this process for all the listed x-values.
Step 3: Plot the obtained points (x, y) on the coordinate plane. Connect the points to create a smooth curve.
After completing the graph, observe where the curve crosses the x-axis (y = 0). These points correspond to the x-intercepts.
In this case, the x-intercepts are at x = 180°, x = 210°, x = 330°, and x = 360°.
Note: The given range (0 < x < 360) implies values within one full rotation (360 degrees) of the sine function.