Finding the Equation of a Quadratic Function (1)

1. When the vertex and the coordinates of another point on the graph are known:
1. Express the quadratic function as .
2. Substitute the coordinates of the given point into the equation to solve for .


2. Examples based on the vertex's coordinates
1. If the vertex is , the equation is
2. If the vertex is , the equation is
3. If the vertex is , the equation is
4. If the vertex is , the equation is


3. When the equation of the axis and the coordinates of two points on the graph are known
1. Express the quadratic function as .
2. Substitute the coordinates of the two points into the equation to solve for and .


4. Examples based on the equation of the axis
1. If the axis is , the equation is
2. If the axis is , the equation is

Finding the Equation of a Quadratic Function (2)

1. When the intersection with the -axis and the coordinates of two other points on the graph are known
1. Express the quadratic function as .
2. Substitute the coordinates of the two points into the equation to solve for and .


2. When the intersections with the -axis and and the coordinates of another point on the graph are known
1. Express the quadratic function as .
2. Substitute the coordinates of the other point into the equation to solve for .

Example 1. A quadratic function represented by a parabola with the vertex at , which intersects -axis at , can be expressed as . Find the value of for the constants , and .

 Solution

Since the vertex is at , we can assume , and by substituting and ,

Since ,
,

Example 2. The graph of the quadratic function () has the same shape with the graph of the quadratic function , and intersects -axis at two points and . Find the value of for the constants , and .

 Solution

Since the shape is same with the graph of , the coefficient of the quadratic term is .
Furthermore its graph intersects -axis at and ,

Hence , , .