The equation for the quadratic graphed in the figure is [tex]-1(x-2)^{2} +3 =0[/tex] (standard form) OR [tex]-x^{2} +4x-1 = 0[/tex] (general form)
To write the equation for the quadratic graphed below,
we will use the standard form of a quadratic function, which is
[tex]y = a(x-h)^{2} +k[/tex]
Where [tex](h, k)[/tex] is the vertex
From the graph, we observe that the vertex is at [tex](2,3)[/tex]
∴ [tex]h = 2[/tex]
and [tex]k =3[/tex]
Now, to determine the value of [tex]a[/tex],
we will pick any suitable point [tex](x, y)[/tex] on the curve.
Using the point [tex](3,2)[/tex]
That is,
[tex]x = 3[/tex] and [tex]y = 2[/tex]
Put these values into the equation [tex]y = a(x-h)^{2} +k[/tex] and solve for [tex]a[/tex]
We get
[tex]2 = a(3-2)^{2} +3[/tex]
Then,
[tex]2 = a(1)^{2} + 3[/tex]
[tex]2 = a +3\\[/tex]
∴ [tex]a =2-3[/tex]
[tex]a =-1[/tex]
Now, we will put the respective values of [tex]a, h, \ and \ k[/tex] into the equation for the standard form of a quadratic function
That is,
[tex]y = a(x-h)^{2} +k[/tex]
Then,
[tex]y = -1(x-2)^{2} +3[/tex]
Therefore the equation is
[tex]-1(x-2)^{2} +3 =0[/tex]
This is the equation in the standard or vertex form.
This equation can be further simplified to give
[tex]y = -1(x-2)^{2} +3[/tex]
[tex]y = -1(x-2)(x-2) +3[/tex]
[tex]y = -1(x^{2} -2x-2x+4) +3[/tex]
[tex]y = -1(x^{2} -4x+4) +3[/tex]
[tex]y = -x^{2} +4x-4+3[/tex]
[tex]y = -x^{2} +4x-1[/tex]
Therefore, the equation is
[tex]-x^{2} +4x-1 = 0[/tex]
This is the general form of the equation.
Hence, the equation for the quadratic graphed in the figure is [tex]-1(x-2)^{2} +3 =0[/tex] (standard form) OR [tex]-x^{2} +4x-1 = 0[/tex] (general form)
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