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Constructing a quadratic function means determining from the given information. Known points lead to a system for the coefficients, known roots lead to factored form, and a known vertex leads to vertex form.
Quadratic functions can be written in three equivalent forms:
These three forms are interrelated and can be converted from one form to another.
Three points with distinct -coordinates determine exactly one polynomial of degree at most . It is a quadratic function only when the resulting coefficient satisfies .
If we have three points , , and , we can substitute these values into the standard equation to get three linear equations with three variables , , and .
Example:
Find the quadratic function that passes through points , , and .
Substitute the coordinate values into the standard equation.
Simplify these equations.
Substitute into the first and third equations.
Simplify and add the two equations to find the value of .
Substitute the value of into the equation to find the value of .
The resulting quadratic function is:
If we know the coordinates of the vertex and another point on the graph, we can use the vertex form to find the value of .
Example:
Find the quadratic function that has a vertex at and passes through the point .
Use the vertex form with and .
Substitute the point to find the value of .
The resulting quadratic function is:
If we know the roots (-intercepts) and of the quadratic function, and an additional point on the curve with , , and , we can use the factored form to determine .
Example:
Find the quadratic function that has roots at and , and passes through the point .
Use the factored form with and .
Substitute the point to find the value of .
The resulting quadratic function is:
The axis of symmetry and the discriminant usually allow several quadratic functions. Another piece of information is needed to select one unique function.
Example:
Find a quadratic function with axis of symmetry and discriminant .
From the axis of symmetry , we know that , so .
From the discriminant , we know that .
Substitute into the discriminant equation.
There are many values of and that satisfy this equation, so the data do not determine a unique function. To obtain one example, choose .
With , , and , the resulting quadratic function is:
Symmetric points can also determine a quadratic function.
Example:
Determine the quadratic function that passes through the points , , and .
Since the points and have the same -value and are at the same distance from the -axis, the curve is symmetric about the -axis. This means the axis of symmetry is , and the vertex is at .
Use the vertex form with and .
Substitute the point to find the value of .
The resulting quadratic function is:
To convert to vertex form :
To convert to factored form over the real numbers, first verify that the discriminant is nonnegative:
Determine the roots of the equation using the formula:
If and the real roots are and , then:
If , the function has no real roots and cannot be written as a product of real linear factors.
To convert to standard form :
By comparing with the standard form, we get:
Find the quadratic function that passes through the points , , and .
Answer:
Substitute the points into the standard equation:
From the second equation, . Substitute into the first equation, then solve:
Substitute into the third equation:
Then:
The quadratic function is:
Determine the quadratic function that has a vertex at and passes through the point .
Answer:
Use the vertex form with and :
Substitute the point :
The quadratic function is:
Determine the quadratic function that has roots at and , and has a maximum value of .
Answer:
Use the factored form with and :
Since the function has a maximum value, .
The vertex is at .
Substitute into the factored form and use the fact that the maximum value is :
The quadratic function is:
Published: . Updated: .
