For the graph of function, find an equation of the line of symmetry and the coordinates of the vertex. Tell whether the value of the function at the vertex if a maximum or a minimum: y=-x^2+4x-7;
You need to convert the given standard form of equation of parabola `y = -x^2 + 4x - 7` , in vertex form` y = a(x - h)^2 + k` , where `(h,k)` represents the vertex of parabola.
You may find the coordinates h and k, using the following formulas, such that:
`h = -b/(2a), k = (4ac - b^2)/(4a)`
Identifying the coefficients a,b,c yields:
`a = -1, b = 4, c = -7 `
`h = -4/(-2) => h = 2`
`k = (-28 - 16)/(-4) => k = 44/4 => k = 11`
Hence, the vertex form of parabola is the following:
`y = -(x - 2)^2 + 11`
You need to find the axis of symmetry of parabola, using the following equation, such that:
`x = -b/(2a) => x = 2`
Hence, evaluating the vertex form of parabola yields `y = -(x - 2)^2 + 11` and evaluating its axis of symmetry yields `x = 2` .
The vertex represents a maximum point (parabola opens downward).