Find the quadratic function y = ax² + bx + c from the graph

Question:

The graph of a quadratic function, y = ax² + bx + c, passes the point A (-1, 0), B (5, 0) and C (0, -10). Find the analytical expression of the graph.

Solution:

Because the graph of the quadratic function, y = ax² + bx + c, intersects the x-axis at point A (-1, 0) and point B (5, 0). So, we use the roots form of the quadratic function. The quadratic function of the graph is: y = a (x - x₁) (x - x₂), in which x₁ and x₂ are two roots of the quadratic equation y = ax² + bx + c = 0. Now substitute the coordinate of point A (-1, 0) and the coordinate of the point B (5, 0) in to the quadratic function, we get the quadratic function of the graph: y = a (x + 1) (x - 5)

Because point C (0, -10) lies on the graph, so, point C satisfy the quadratic function. Now we substitute the coordinate of the point C (0, -10) into the quadratic function y = a (x + 1) (x -5)

y = a (x + 1) (x -5)

-10 = a (1) (-5)

-10 = -5a

a = 2

Thus, the quadratic function of the graph is y = 2 (x + 1) (x - 5). Now we write it in the general form

y = 2 (x + 1) (x - 5)

y = 2 (x² - 5x + x - 5)

y = 2 (x² - 4x - 5)

y = 2x² - 8x - 10

Therefore, the quadratic function of the graph is y = 2x² - 8x - 10. Watch the video for more details.