Can anyone explain how the values of "a", "b" and "c" affect the graph of the quadratic function, specifically?

Question

Can anyone explain how the values of "a", "b" and "c" affect the graph of the quadratic function, specifically?

Answer 1

If your equation is in vertex form, y = a(x-h)^2+k, it's easy to understand what varying a, h, and k do to the graph. |a| makes the graph skinny or fat, a > 0 opens up, <h,k> translate (shift) the graph in x and y direction.

 

To see what happens for the standard form let's multiply the vertex form out and compare the two:

 

y = a(x-h)^2+k

 

y = a(x^2-2hx+h^2)+k

 

y = ax^2-2ahx+ah^2+k

 

Now compare to y = ax^2+bx+c:

 

a=a, b=-2ah, c=ah^2+k

 

h=-b/(2a), k=c-ah^2=c-a(-b/2a)^2=c-(b^2)/4a

 

Putting these values back into the vertex form:

 

y = a(x+b/(2a))^2+c-(b^2)/4a

 

While a and b contribute to the x shift, all three contribute to the y shift.

 

Play with it here:

http://www.geogebratube.org/student/m4538

Answer 2

Hi Ness;

f(x)=ax²+bx+c

 

If a is positive, the parabola is upward.

If a is negative, the parabola is downward.

 

The y-intercept, value of y when x equals zero, is c, (0,c).

 

The vertex is -b/2a.  This is easy to understand.  The vertex is the point at which the change of slope is zero.  Let's take the first derivative of the equation and set it equal to zero...

0=2ax+b

-b=2ax

-b/2a=x