 In this video, we provide the solution to question number 16 from practice exam number one for math 1050 We have to graph the function f of x equals negative 2 times the absolute value of x minus 2 plus 3 And there's some instructions that go along the way here. How are we going to graph this? First of all, we need to indicate all of the transformations that went upon Y equals the absolute value of x the standard absolute value to get us to where we are here And so that's my first recommendation. What are the transformations going on here? So notice we have this coefficient of negative 2 in front of the absolute value The fact that it's negative means the absolute value is reflected across the x-axis The fact we have a 2 there means it was vertically stretched by a factor of 2. So let's illustrate that I Illustrated let's let's write it down. So we actually have a reflection reflection across The x-axis So that's the first transformation The second transformation is that it was vertically stretched Vertically stretched by a factor By a factor of 2. All right What does this negative 2 inside of the absolute value mean there if we're inside the absolute value because that's this That was the original basic function Then we're inside of the horizontal zone And so x minus 2 inside the absolute value suggests there is a Horizontal shift, but it's a shift in the positive direction. And so we have in fact a shift right Shift right by 2 like there and then lastly we have this plus 3 notice it's outside of the horizontal zone So this is a vertical transformation. So the plus 3 is in fact going to be a shift upward shift up By a factor of 3 in order to get full credit on this question You must indicate all of these transformations these four transformations In play right there now for our convenience the graph of the absolute values provided to us This is our function f right here because maybe we don't know it. That's okay Not a big deal. This question is asking about the transformations And so now we want to graph it based upon these transformations It also asks us to indicate three points on the graph And so what I would suggest is start off with three points on the original graph So we have the origin zero zero we have the point one one and we have the other point Want negative one one excuse me. So pay attention to what happens to these points as we transform them If you take the origin and you reflect it across the x-axis that doesn't do anything to it The y-coordinate becomes negative zero, which is still zero if you stretch it vertically by a factor of two That would stretch most y-coordance, but zero times two is still zero. So it didn't do anything to that as well So any point on the x-axis is not affected by these vertical reflections and stretches If you move things to the right by two that does affect it You're gonna move over by two and then you shift it up by three like so you see that the Vertex of the absolute value moves over to this point right here and do label it We get the point 2 comma negative 2 comma 3 like so Let's look at some of these other points take one one for example if we reflect it across the x-axis that moves it down here Moves it down here to be one negative one. We're they gonna stretch it vertically by factor of two That means we're gonna times it's y-coordinate by two So that's gonna move it over here to be one negative two We didn't have to move it to to the right So that puts us here and then we have to move it three up one two and then three like so In which case then that gives us this point right here, which would be the point three comma one Like so let's play this game one more time Negative one one when we reflect across the x-axis gives us this point here Vertically stretching it by a factor of two gives us this point right here. We're at negative one negative two We move two to the right. We move three upwards So actually at the point one one again, but for a different reason So we get one one And then the rest of the graph from these three points It's absolute value looks like a V it should just look like the original graph although There's some distortion that happened because of the reflection and then stretching there So if you have a straight edge use it if not not a big deal connect the dots right here We get that connect the dots right there Like so and so in the graph We get this red graph as an F. I think I labeled earlier the blue graph as F. That's not true Sorry, this the blue graph was the absolute value of x Which was given below we now have the graph of F here on the screen It's a good practice To draw little arrows at the end to suggest that the function does extend beyond it's not a finite domain It goes on and on and on and on but that's what we need to do to get full credit on this question We need to list our transformations We need to have a correct graph that utilize those transformations And we do need to indicate at least three points on the graph So we can see very quickly Assuming we have a hard time drawing pictures. We need the group the points to help us indicate What exactly it was we were trying to graph