The graph of a quadratic function is shown above.It has a vertex at (3,1)

Question

and passes through the point (0,−8). Find the quadratic function.?

Answer 1

Vertex form of parabola is y = a(x - h)² + k

Substitute (x ,y) = (0, - 8) and (h, k ) = (3, 1) in y = a(x - h)² + k.

- 8 = a(0 - 3)² + 1

Solve for a.

- 8 = 9a + 1

9a = - 9

a = - 1

Substitute a  and (h, k) values in y = a(x - h)² + k.

y = - 1(x - 3)² + 1

y = - 1(x² + 9 - 6x) + 1

y = - x² - 9 + 6x + 1

Required equation y = - x² + 6x - 8

Now it is in the form of quadratic function y = ax² + bx + c.

a = - 1, b = 6  and c = - 8.

Comments

The equation is y = - x² + 6x - 8.

Make the table of values to find ordered pairs that satisfy the equation.

Choose random values for x and find the corresponding values for y.

x	y = - x2 + 6x - 8	(x, y)
1	y = - (1)2 + 6(1) - 8 = - 3	(1, -3)
2	y = - (2)2 + 6(2) - 8 = 0	(2, 0)
4	y = - (4)2 + 6(4) - 8 = 0	(4, 0)
5	y = - (5)2 + 6(5) - 8 = - 3	(5, - 3)
6	y = - (6)2 + 6(6) - 8 = - 8	(6, 8)

Graph :

Draw the coordinate plane.

Plot the points found in the table.

Connect the plotted points with smooth curve.

Observe the graph, The vertex of parabola is (3, 1) and it passes through (0, - 8).