Question Video: Determining the Solution Set of a Quadratic Equation Using Its Vertex and Leading Coefficient Mathematics • Third Year of Preparatory School

Video Transcript

True or False: If the point negative two, negative five is the vertex of the graph of the quadratic function 𝑓 of 𝑥 is equal to 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎 is a negative number, then the solution set of the equation 𝑓 of 𝑥 is equal to zero is the empty set.

In this question, we’re given some information about a quadratic function 𝑓 of 𝑥. We’re told that the value of 𝑎, its leading coefficient, is a negative number. We’re also told that the point with coordinates negative two, negative five is the vertex of the graph of this quadratic function. We need to use this information to determine if the solution set of the equation 𝑓 of 𝑥 is equal to zero is the empty set.

And to do this, let’s start by recalling what we mean by the solution set of an equation. It’s the set of all solutions to the equation. So, for the equation 𝑓 of 𝑥 is equal to zero, it’s the set of all values of 𝑥 such that 𝑓 evaluated at 𝑥 is zero. And we need to determine if it’s true whether the solution set to this equation is the empty set, which means it has no solutions.

There’s several different ways we could go about doing this. For example, we could try and find the solutions algebraically. However, this is quite difficult. So, instead, because we’re given the sign of the leading coefficient of our graph and the coordinates of its vertex, we’re going to do this graphically. Let’s start by sketching a graph of the function. And to do this, we can recall all quadratic curves have a parabolic shape. And in particular, there’s two orientations for a parabola which are determined by the sign of the leading coefficient.

If the leading coefficient is negative, then we say that the parabola opens downwards. However, if the leading coefficient is positive, then we say that the parabola opens upwards. And in our case, we’re told that the sign of the leading coefficient 𝑎 is negative. So the shape of our parabola will open downwards. We can also recall that the turning point of these parabolas is called the vertex of the parabola. If the parabola opens downwards, then the 𝑦-coordinate of the vertex tells us the maximum output of the function. And if the parabola opens upwards, then the 𝑦-coordinate of the vertex tells us the minimum output of the function. And it’s also worth noting all parabolas are symmetric through the vertical line through their vertex.

And now we can sketch the graph of our parabola. We’ll start by drawing a pair of coordinate axes and marking the coordinates of the vertex negative two, negative five. We then want to sketch a parabola with this point as the vertex which opens downwards. For example, we might get the following. However, it’s worth noting we don’t know how narrow or wide this parabola will be. For example, we might have a wider parabola which still has the point with coordinates negative two, negative five as its vertex and opens downwards. And we might have a more narrow parabola, such as the following. We don’t know the exact shape of this curve.

However, we can notice something interesting all of the parabolas have in common. They all have the vertex negative two, negative five. And now since our parabola opens downwards, we know the 𝑦-coordinate of the vertex tells us the maximum output of the function. Its maximum output is negative five, and this occurs when 𝑥 is equal to negative two. But if the maximum output of the function is negative five, the function will never output zero. So it has no solutions. So the equation has no solutions. So its solution set is the empty set.

And it’s worth noting this is equivalent to saying that the graph of the function has no 𝑥-intercepts because an 𝑥-intercept would be the values of 𝑥 where the function outputs a value of zero. So we could’ve equivalently determined that the statement is true by noting none of our sketches will pass through the 𝑥-axis. In either case, we were able to show if the point negative two, negative five is the vertex of the graph of a quadratic function with negative leading coefficient, then it’s true the solution set of the equation 𝑓 of 𝑥 is equal to zero must be the empty set.