Grades 11-12 Video Solutions 2011-11&12 Video Solutions 2011 problem15

Video Transcription

Question number 15. How many of the following eight functions have their graphs contained in the graph, part of which is shown below? Let's begin by  analyzing the square root function. That function is non-negative on its domain, which is the non-negative numbers, and it's always increasing, so part of it is  shown here in quadrant one. That's the square root of x. Its reflection about the x-axis would be in quadrant four, and that is the negative square root of x.  That is right here. Then its reflection about the y-axis in quadrant two would be given by the square root of negative x, and the reflection of that function  would belong to quadrant three, and the formula here is the negative square root of negative x. So we have four functions represented. Then we can combine the  ones in quadrant one and two and simplify the notation a little bit. We recall that the absolute value function is defined as x for x non-negative and  negative x for x negative, so we can simplify and combine these, so together the functions appearing in the first and second quadrant are the square root of  the absolute value of x, and upon removing the absolute value, we have to consider two of these cases. Similarly, the functions in quadrants three and four  would be the reflection about the x-axis, and that's the negative of the square root of the absolute value of x, and we have two functions that are missing.  y is equal to x squared, and y is equal to the negative x squared, the reflection of that about the x-axis, and we can find some points on these functions. For  example, on x squared, the point 1, 1 would belong to that function, and if we plug in 2, we would obtain a 4, which is clearly not here. Similarly, the  reflection of 2, 4 about the x-axis would give us 2, negative 4, which is also not present, so we don't have these two functions. All the other ones are  represented, and there are exactly six of them, so the answer is D.

Video Summary

The video analyzes functions whose graphs are part of a larger graph depicted in the video. It discusses the square root function and its reflections across the axes, involving four functions: \(\sqrt{x}\), \(-\sqrt{x}\), \(\sqrt{-x}\), and \(-\sqrt{-x}\). Using absolute value, these are grouped into \(\sqrt{|x|}\) and \(-\sqrt{|x|}\), covering quadrants one through four. The quadratic functions \(y = x^2\) and \(y = -x^2\) are missing from the graph, indicated by their points not appearing. In total, six of eight functions are represented, with D being the correct answer.

Keywords

square root function

reflections

quadrants

quadratic functions

graph analysis