Worksheet on relating the graph of a quadratic function to its discriminant,
and its equation:
Consider the function f(x) = ax2+ bx + c, where a,b,c and a 0.
If a > 0, the graph of f(x) , called a parabola, opens up. (Think of a smile). We say
the curve is concave up. The parabola in fig 1 is concave up, a > 0.
Fig 1
If a <0, the parabola, opens down. (Think of a frown). We say the curve is
concave down. The parabola in fig 2 is concave down, a< 0.
Fig 2
The value of c is the y-intercept (where the graph crosses the y-axis). c is a real
number, so c>0, or c<0 or c = 0. In figure 1, c=1, so c >0. In figure 2, c = -4, so
c <0. You sketch a graph of a parabola where c = 0.
When we use the quadratic formula to solve quadratic equations, we need to
calculate the number inside the square root. This number, b 2 - 4ac, has a special
name. It is called the discriminant. Let D = b2 - 4ac. Then D is a real number,
and therefore the value of D has three possibilities:
Case 1: D > 0. Then the number inside the square root is positive, and there are
2 real solutions to the equation ax2+ bx + c = 0. This means the graph of f(x) =
ax2 + bx + c will cross the x-axis in 2 places.
In figure 3, the graph crosses the x-axis at x = 1 and x = -3, (crosses in 2 places)
so D > 0.
Fig
3
Case 2: D < 0. Then the number inside the square root is negative, and there are
2 complex solutions to the equation ax2+ bx + c = 0. This means the graph of f(x)
= ax2 + bx + c will never cross the x-axis .
In figure 1, the graph never crosses the x-axis, so D < 0.
Fig 1
Case 3: D= 0. Then the number inside the square root is 0, and there is only one
real solution to the equation ax2+ bx + c = 0. This means the graph of
f(x) = ax2 + bx + c will touch the x-axis in only one point, which is the
b  0
b
location of the vertex of the parabola, x 
 .
2a
2a
In figure 2, the graph touches the x-axis at the vertex, x =2, so D = 0.
Fig2
By inspection, for each of the following graphs determine the following:
 a >0 or a < 0
 D > 0, D < 0 or D = 0
Example: For the graph depicted in figure 4, we have
 c > 0, c <0, or c = 0
a< 0, D> 0 and c > 0
Fig 4
#2
a
D
c
#1.
a
D
c
#4
a
D
c
#5
a
D
c
#3
a
D
c
#6
a
D
c