Given: y = -x^2 + 3y=−x2+3
With the equation in the form: Ax^2 + Bx + C = 0Ax2+Bx+C=0,
the vertex is at (-B/(2A), f(-B/(2A)))(−B2A,f(−B2A)),
the axis of symmetry is x = -B/(2A)x=−B2A
If the coefficient A < 0A<0, the vertex is a maximum
If the coefficient A > 0A>0, the vertex is a minimum
For the given equation:
-B/(2A) = 0/-2 = 0−B2A=0−2=0
f(0) = -(0)^2 + 3 = 3f(0)=−(0)2+3=3
vertex (0, 3)(0,3) is a maximum; axis of symmetry: x = 0x=0
color(blue) ("Find x-intercepts")Find x-intercepts by setting y = 0y=0:
0 = -x^2 + 30=−x2+3
-3 = -x^2−3=−x2
x^2 = 3 => x = +- sqrt(3)x2=3⇒x=±√3
xx-intercepts: (-sqrt(3), 0), (sqrt(3), 0)(−√3,0),(√3,0)
color(red) ("Find y-intercept")Find y-intercept by setting x = 0x=0:
y = -(0)^2 + 3 => y = 3y=−(0)2+3⇒y=3
yy-intercept: (0, 3)(0,3), which is the vertex