The domain of the graph of a quadratic function with vertex (1, -3) is all real numbers and range belongs to the number less then equal to -3.
What is the vertex form of quadratic?
Vertex form of quadratic equation, is used to find the coordinate of vertex points at which the quadratic crosses its symmetry.
The standard equation of the vertex form of quadratic is given as,
[tex]y=a(x-h)^2+k[/tex]
Here, (h, k) is the vertex point.
The graph of a quadratic function with vertex (1, -3) is shown in the figure below. Put the value in the above equation,
[tex]y=a(x-1)^2+(-3)\\y=a(x-1)^2-3[/tex]
One point of this function is passes from (0,-4) as shown in graph. Put this value in above expression to solve it for a,
[tex]-4=a(0-1)^2-3\\-4+3=a\\-1=a\\a=-1[/tex]
Put the value of a, we get,
[tex]y=-1(x-1)^2-3\\y=-(x-1)^2-3\\y=-x^2-1+2x-3\\y=-x^2+2x-4[/tex]
The domain of the above function is all real number as there is no undefined point and nor domain constraint. Thus, the domain is,
[tex](-\infty < x < \infty)[/tex]
In the above function, the coefficient of the term with the highest degree is -1. Put this value as,
[tex]y=-(1)^2+2(1)-4\\y=-1+2-4\\y=-3[/tex]
Hence, the range of the function,
[tex]f(x)\le-3[/tex]
Thus, the domain of the graph of a quadratic function with vertex (1, -3) is all real numbers and range belongs to the number less then equal to -3.
Learn more about the vertex form here;
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