Sections 2.1 quadric functions

Question

Find the quadratic functions y = f(x) that has a given vertex and whose graph passes through the given point: Write the functions in standard form: F(x) = a(x-h)²+k

1.) Vertex (2,5) passing through (3,7) ------------------------> answer: y = 2 (x-2)² +5

Answer 1

In this form of the equation the coordinates of the vertex are at (h, k)

So we know that h = 2 and k = 5

We also know that the lines passes through the point (3, 7)

So when x =3 then y = 7.

Now we have only one unknown left. It is "a"

Putting all our values into

F(x) = a(x-h)²+k remembering that F(x) is the same as y

7 = a (3-2)² + 5

So 7 = a (1)² + 5

and 7 = a + 5

So a = 2

Replacing a, h, and k in the original equation gives

F(x) = 2 (x - 2)² + 5

This is the correct answer.

Answer 2

Start with the standard for of the equation and the vertex. Use the vertex to populate h and k.

F(x) = a(x-h)²+k Vertex: (2,5)

F(x) = a(x-5)²+2 Now use the point. (3,7) and plug those in for x and y.

7= a(3-5)²+2 Now solve for a

7= a(-2)²+2 Simplify the parenthesis

7= 4a+2 Subtract 2 from both sides

5=4a Divide both sides by 4

5/4=a

Now, bring everything together for the equation

F(x) = 5/4(x-5)²+2