Find a formula for the quadratic function

Question

whose graph has vertex (-1,-5) and y-intercept -2

 

f(x)=

Answer 1

Okay, so we know that a quadratic function f has a standard form, f(x) = ax² + bx + c.  However, we also have the form f(x) = a(x - h)² + k, where (h,k) is the vertex of the parabola.  

 

So from our given vertex of (-1,-5) we have f(x) = a(x + 1)² - 5.

 

Now we need to solve for a.  To do so, we need to note that the y-intercept is -2.  

This means that f(0) = -2, because the y-intercept occurs where x = 0.  

 

But we also know from the formula that f(0) = a(0 + 1)² - 5 = a - 5.

 

So now we set the two equations equal to each other:

 

Thus f(0) = a - 5 = -2.  

 

Solving for a, a = -2 + 5 = 3.

 

Now we know that f(x) = 3(x + 1)² - 5.

 

From here, you should be able to put this formula into standard form if necessary.  Try it anyway for practice!

 

Does this make sense?  Feel free to ask questions if needed.  

 

Also, for more practice, try the following:

Find a formula for the quadratic function whose graph has vertex (2,-7) and y-intercept 5.