Question

Graph the equation y = -x² + 4x + 5 on the accompanying set of axes. You must

plot 5 points including the roots and the vertex. Using the graph, determine the
vertex of the parabola.

Answer 1

see attached

Explanation:

You can find the x-intercepts by factoring the equation:

y = -(x² -4x -5) = -(x -5)(x +1)

The x-intercepts (roots) are the values of x that make the factors zero:

x -5 = 0 ⇒ x = 5

x +1 = 0 ⇒ x = -1

The axis of symmetry is halfway between, at x=(5-1)/2 = 2.

For purposes of graphing, it is useful to use values of x that are on the axis of symmetry, and 1 unit either side.

For x=2, y = -(2 -5)(2 +1) = 9

For x=1, y = -1² +4·1 +5 = 8

For x=0, y = 0 +0 +5 = 5

The graph is symmetrical about the axis of symmetry, so you now have the points ...

(-1, 0), (0, 5), (1, 8), (2, 9), (3, 8), (4, 5), and (5, 0)

The vertex is (2, 9).

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Additional comment

The values 1 unit either side of the vertex differ from the vertex value by the amount of the leading coefficient. Here, that is -1, so the y-values for x=2±1 are 9 +(-1) = 8. This is useful to remember when you are writing equations from a graph.