Graph each function. Determine the largest open intervals of the ...

Question

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=1/2(x-2)^2+4

Accepted Answer

Hello, today we're gonna be identifying the regions of increasing and decreasing for the following function. Now the function given to us is 1/5 times the quantity X minus five squared plus three. Now the way that we can easily approach this is to use the method of graphing using transformations. Now notice that the highest exponent for X is two. So that tells us that the parent function for this equation is going to be Y is equal to X squared. And this represents a parabola whose vertex is located at the origin. So what we need to do now is we need to go uh go ahead and list out all the possible transformations for this graph. While the first transformation is going to be the quantity X minus five. See this is in the form of X minus H. And since this is the standard form of a horizontal shift, this is going to be where H is equal to five. So the first transformation we're going to have is a horizontal shift where we are shifting the graph five units to the right. Next, the second transformation is going to be plus three. So what this means is that for every value of X that we're plugging in, we're increasing the Y value by three. So the plus three is also represented as plus K and that is going to give us a vertical shift of three units up. And finally, the last transformation is going to be with 1/5. See this represents a coefficient in front of the graph. And so for every value of X that you plug into this equation, you're gonna multiply that value by the coefficient of 1/5. Now, because 1/5 is less than one that is going to mean that we're gonna horizontally stretch the graph. So the third transformation is going to be a stretch of the graph horizontally. So let's go ahead and first apply the first two transformations. The current vertex is at zero comma zero. If we move it five units to the right and five units up the location of the new vertex is going to be at five comma three. Now, we could just roughly draw the equation of the parabola. But let's go ahead and look for the Y intercept next. In order to get the Y intercept, we're going to go ahead and plug in X is equal to zero because the Y intercept represents the region of the graph that intercepts the Y axis when X is equal to zero. So plugging in X is equal to zero is going to give us 1/ times zero minus five squared plus three. Now negative five squared is going to give us positive 25. So we have 1/5 times 25 plus three, one or five times 25 is going to leave us with five. So we have five plus three and five plus three is going to equal to eight. So when X is equal to zero, we get a Y value of eight And that tells us that the Y intercept is going to be at zero comma eight. So we can go ahead and roughly label that point on our current graph. And that means that the rough shape of the new transformation is going to be this graph right here. Now notice that the vertex starts when X is equal to five, we can go ahead and use this to find the region of increasing and decreasing. The reason for the spin is because as the graph approaches the location of five, when approaches from negative infinity to positive infinity, the graph is going to be decreasing to that point. So that tells us that the region of decreasing is going to be from negative infinity to positive five. Now we need to find the region of increasing. Now what this means is that at the 0.5 as you will continue to travel to the right, this graph is now going to travel to positive infinity or in other words, it's going to continue to increase. So that means that the region of increasing is going to start at five and continue to positive infinity. So now that we have the rough shape of the graph and we have the region of increasing and decreasing, we just need to find the graph that accurately represents this information. Well, that graph is going to be option B we have the vertex at five comma three and we have the Y intercept at zero comma eight. Furthermore, we have the region of decreasing from negative eight, from negative infinity comma five and the region of increasing to five comma infinity. And with that being said, the answer for this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

My Courses

Chemistry

Biology

Math

Physics

Business

Social Sciences

Programming

Product & Marketing

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=1/2(x-2)^2+4