Find the vertex of the given parabola. Then, determine if the vertex lies above or below the given line.
Two points of intersection
Practice makes perfect
We want to determine the number of points of intersection of the given system of equations. We will do it without actually graphing or solving the system.
y=- 2(x-1)^2-3 & (I) 5x+2y+10=0 & (II)
Note that the first equation of the system is a quadratic equation and the second one is a linear equation.
We want to find the number of points in which the graphs of the given equations — a parabola and a line — intersect. When it comes to a line and a parabola, we have three possibilities.
Let's begin by identifying the vertex of the given parabola.
It is important to note that we do not need to graph the parabola to identify the desired information. Let's compare the general formula for the graphing form to our Equation (I).
General Formula:y=& a(x- h)^2 + k
Equation:y=& - 2(x- 1)^2+(-3)
The vertex of a quadratic function written in graphing form is the point ( h,k). Thus, the vertex of this parabola is ( 1,- 3). Let's also look at the value of a.
Recall that if a>0 the parabola opens upwards. Conversely, if a<0 the parabola opens downwards.
In the given function we have a= - 2, which is less than 0. Thus, the parabola opens downwards.
Relationship Between the Vertex and the Line
Consider the relationship between the position of the vertex and the line. Let's determine if the vertex of the parabola lies above or below the line. To do it we will find the y -coordinate of the point on the line with the x -coordinate 1, which is the x-coordinate of the vertex.
The y -coordinate of the vertex is - 3. Since - 7.5 is less than - 3, the vertex lies above the line. We also know that the parabola opens downwards. It means that the quadratic function and the linear function must have two points of intersections.
