Grades 11-12 Video Solutions 2013-Levels 11&12 Video Solutions 2013 problem15

Video Transcription

Question number 15, given as a function w of x equal to quantity a minus x times quantity b minus x squared where a is less than b. Its graph is in one of the following figures.  In which one?  We note that the numbers a and b are x-intercepts of the function w with a occurring before b and so let's study the behavior of the graph  at each intercept to pick the correct answer.  Now since we're looking at the function w which is a polynomial we note that its behavior is determined  by the factor from which an intercept comes from if we look near the values of that intercept. So we can say that near x equal to a, w of x behaves  like the factor a minus x and this is a line. And a line unless it's horizontal intercepts the x-axis by passing through it completely.  And similarly near x is equal to b the function w looks like the factor b minus x quantity squared  and this is a parabola with its vertex at the x-axis. So what we need in terms of our intercepts is the first intercept must pass through the x-axis  and the second intercept must bounce off or touch the x-axis like a parabola would.  So that's the behavior in a and that's also the behavior in d  and now we need to decide between those two.  One way to do that is to study the end behavior of the graph.  So here in a when x is large the graph decreases and in d when x is large the graph increases.  So we decide between those two behaviors by looking at the function w again and multiplying it out.  And if we do that we have the leading term which will be the power of x and with a minus sign  and the remaining terms are of lower degree.  So we only need to worry about the first term  with the highest power of x which will dominate the behavior of the function for large values and we note that since negative x cubed would approach negative infinity  as x approaches positive infinity the same is true of the function w. wx approaches negative infinity as x approaches positive infinity.  So that's exactly the behavior we have in the graph with choice a and so that's our candidate solution.

Video Summary

The function \( w(x) = (a - x)(b - x)^2 \), with \( a < b \), has \( a \) and \( b \) as its x-intercepts. Near \( x = a \), \( w(x) \) behaves like a line and crosses the x-axis, while near \( x = b \), it behaves like a parabola, touching the x-axis. Thus, the graph first crosses and then touches the axis at intercepts. To select the correct graph, we analyzed end behavior: as \( x \to \infty \), \( w(x) \) decreases to \(-\infty\), which corresponds to graph choice \( a \).

Keywords

function behavior

x-intercepts

end behavior

graph analysis

parabola