Question

A quadratic function has a vertex of (1,5) and the point (-1,13) lies on the function. Determine the function that contains this vertex and point in Standard Form.

Answer 1

To determine the quadratic function, the vertex form is used to find the value of 'a,' and then the function is expressed in Standard Form, resulting in f(x) = 2x² - 4x + 7.

Step-by-step explanation:

The problem requires us to determine a quadratic function in Standard Form with a vertex at (1,5) and a point (-1,13) that lies on the function. A quadratic function in the Standard Form is written as f(x) = ax² + bx + c. Using the vertex form of a quadratic function f(x) = a(x-h)² + k, where (h,k) is the vertex, we can substitute the vertex to get f(x) = a(x-1)² + 5. Afterward, we plug in the point (-1,13) to solve for the value of a:

f(-1) = a((-1)-1)² + 5 → 13 = a(4) + 5 → a = 2

Now that we've determined a = 2, we express the function in Standard Form:

f(x) = 2(x-1)² + 5 → f(x) = 2(x² - 2x + 1) + 5 → f(x) = 2x² - 4x + 7