Determine the simplest form of an equation for each graph. Cho...

Question

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.

Accepted Answer

Welcome back. I am so glad you're here. We are asked to write an equation that represents the graph shown. The equation should be in simplest form B is greater than zero and phase shift is not included. Then we're given a graph, we have a vertical Y axis, a horizontal X axis, they come together at the origin. The range for what's drawn for our Y axis is from negative one to positive five. And that's in increments of one. And the domain for what's drawn for our X axis is from 0 to 5 pi divided by two. And that's in increments of pi divided by two. On this graph, we have a function line and the function line is kind of a smooth wave. It begins at the point drawing an xy table writing down these points begins at the 0.03 and then it curves smoothly down to the next point at pi divided by two, two. And then it begins smoothly curving upward, passes through the point pi three continues heading upward to the 0.3 pi divided by 24 and then curves downward and ends at the 0.2 pi three all right. So that's what we're given. And now we need to figure out an equation that represents that graph. So when we see a smooth wave curve like that, we know that we're either dealing with a cosine curve or a sine curve. But which one? Well, we want to look because we know that there's no phase shift at how our function line interacts with the Y axis for the cosine function. If it hasn't been reflected over the X axis, and again, no phase shift, then it's going to pass through the Y axis at the 0.01. And that is going to be a maximum for the sine curve if it hasn't but had vertical shifting. If there's no phase shift, it's midline is going to pass through the origin at 00. Now, as we said, our function line interacts with the Y axis at the 0.03. So clearly, there's some vertical shifting going on and that's going to be important in a little bit. But what's important here is is that a max a min or a midline where it interacts with the Y axis. So is that a maximum, a minimum or a midline? And we can see that that is the midpoint of our curve at 03, it can go down for the Y values to two and it can go up for the Y values to four. So when it interacts at three, that's right in the middle So we're dealing with a sign curve. So we recall from previous lessons, the equation for a sine curve is given with Y equals C plus A sign of B multiplied by the quantity of X minus D. And that is our standard sine curve. And we just need to determine what our A, our B, our C and our dr plug those in and then we'll have our function. Now, a few things to note here. So the A is going to deal with our amplitude and whether it's reflection reflected over the X axis, our bee is going to deal with our period C is going to deal with any vertical shifting up or down. And D is going to deal with any phase shifting left or right. But when we were told in our instructions, there is no phase shift, we know our D value is going to be equal to zero. There is no phase shift. So let's figure out what A B and C are. Well, one way that can be helpful to think about this is to actually compare it to the typical sine curve. So let's draw that real quick. A standard sine curve that has an, a value of positive one A B value of positive one and AC value of zero would be going through these points. We recall from previous lessons that it's going to go through at the origin at 00 and then it's going to go to pi divided by 21. And then it'll curve back down to pi zero and then it'll continue going down to three pi divided by two negative one. And then its period will end when it returns to the Y value of zero. But at the X value of two pi so two pi zero, we can connect those as smoothly as possible. And there we have our typical sine curve. So what's interesting comparing our given function to the typical sine curve is we can see that normally the sine curve has that midline at the Y axis and then it starts going up, but ours has its midline at the Y axis and then it starts going down. That means it has been reflected over the X axis. So our a value that tells us the amplitude and if there's reflection over the X axis that is going to be negative because there's reflection over the X axis. Now we also need to figure out the period or excuse me, the amplitude, we recall from previous lessons. The amplitude can be found by taking one half and multiplying that by the distance between our maximum and our minimum, we already identified the maximum for our Y values is four and the minimum for our Y values is two. So excuse me, it's one half multiplied by the absolute value of that distance. So one half multiplied by. So the distance we take our maximum four minus our minimum two. And so we're taking one half multiplied by the absolute value of four minus two is two. So one half multiplied by two, which is going to be one. So our amplitude is one. So our, a value is going to be a negative one. Our B value is the period. We recall from previous lessons that a period for a sine curve is going to be found by taking two pi and dividing that by B. And so we can see here that the length that it takes to go through one complete period is two pi it's our standard two pi so our B is going to be equal to one. And then our C tells us vertical shifting. Well, yes, we can see that it has shifted upwards. So it's going to be a positive C value and it's gone up. If we look at our midline, our midline has gone up three units. So it's going to be a positive three for our C value. So you want to compare those midline values. So where it's at zero for X pi for X and two pi for X, they've all gone up three units. So we have our ABC and D now we can fill those in. So we have the Y equals C is a positive three plus A. Well, A is negative one. So we can put minus, we don't have to write the one because it's being multiplied. So minus sign of and then B is 11 that's being multiplied. We don't need to write it X minus DD is zero. We don't need to write X minus zero. So it's just sine X. So we have Y equals three minus sine X. And there is the equation that represents our graph. Well done. We'll catch you on the next one.

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.
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Key Concepts

General Form of Trigonometric Functions

Amplitude and Period

Phase Shift and Horizontal Translation

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