Grades 11-12 Video Solutions 2011-11&12 Video Solutions 2011 problem18

Video Transcription

Question number 18. On a plane with a rectangular coordinate system, the point A with coordinates 1, negative 10 is marked on the parabola. Which of the  following inequalities is not necessarily true? Let's go through each of the possibilities and decide if it is  possible or not to have each inequality. Looking at the graph, we can see that the parabola opens up. Or another way to say that is that the graph is concave up at  the point A. And so in that case, we see that A, the first coefficient, the leading coefficient has to be positive. And so answer A is possible. If we then  substitute x is equal to 1 into our equation, what we obtain is y, the value at 1 is A times 1 squared plus B times 1 plus C, which is exactly A plus B plus C.  And the value at 1 is equal to negative 10. So A plus B plus C, that is a negative quantity. So C is possible. Then we can find the vertex of this parabola.  Since it's concave up, the vertex lies to the right of the point A.  and a formula for the vertex is the x-coordinate of the vertex is given by negative b over 2a so here we see that negative b over 2a has to be bigger  than 1 and another way to state that is that negative b is bigger than 2a but a being positive means that negative b is also positive and if negative b is  positive then b has to be negative so choice B is possible and then finally we can say more about the concavity condition the parabola being concave up  at 1 comma negative 10 means that this point a lies below the x-axis and the parabola opens up so it will have to intersect the x-axis at two different  points so 2x intercepts and another condition for that is from the quadratic formula we look inside the radical and that's called the discriminant of the  quadratic equation and it is exactly b squared minus 4ac the condition here is that it is strictly bigger than zero so b squared is bigger than 4ac so we have  checked that condition D is also possible which leaves us with E

Video Summary

The video transcript discusses determining which inequality is not necessarily true for a given point on a parabola. It states that the parabola is concave up, leading to a positive leading coefficient. From substituting \( x = 1 \), the equation yields \( A + B + C = -10 \), confirming it's negative. For a concave-up parabola, the vertex is to the right of point A, leading to \( -b > 2a \), implying \( b \) is negative. The quadratic's discriminant, \( b^2 - 4ac > 0 \), confirms 2 x-intercepts. Thus, options A, B, C, and D are validated, leaving option E as potentially false.

Keywords

parabola

inequality

concave up

discriminant

vertex