c The graph of a quadratic function is a parabola.
D
d When the discriminant of a quadratic function is less than zero, the function has just an imaginary solutions.
E
e The x-intercepts of a quadratic function represents the solutions of the corresponding quadratic equation.
A
aNumber of roots: 1
Example Graph:
B
bNumber of real roots: 0
Example Graph:
C
cNumber of roots: 2
Example Graph:
D
dNumber of real roots: 0
Example Graph:
E
eNumber of real roots: 2 or 1
Example Graph:
Practice makes perfect
a Recall that when the discriminant of a quadratic function is given by the quantity b-4ac. When it is zero, the function has just one zero. This can only happen if the vertex lies on the x-axis.
Notice that this is just an example graph, as there are infinitely many parabolas satisfying this condition. In all these cases, the corresponding quadratic equation will have one root.
b For the function f(x) to not have zeroes, we need to graph a parabola with no x-intercepts. For this to happen, if the parabola open upwards the vertex has to be above the x-axis. If the parabola open downwards the vertex has to be below the x-axis.
Notice that this is just an example graph, as there are infinitely many parabolas satisfying these conditions. In all these cases, the corresponding quadratic equation will have no real roots.
c We need to graph a quadratic function with two different zeroes, f(a)=0 and f(b)-0; a≠0. Its graph will be a parabola with two different x-intercepts. To obtain this we need a parabola whose vertex is below the x-axis if it opens upwards and above the x-axis if it opens downwards.
Notice that this is just an example graph, as there are infinitely many parabolas satisfying these conditions. In all these cases, the corresponding quadratic equation will have two roots.
d Recall that when the discriminant of an equation is less than zero we have two imaginary solutions. As the solutions are not real numbers, the graph has no x-intercepts. The graph will be of the same form as that in Part B.
Notice that this is just an example graph, as there are infinitely many parabolas satisfying this condition. In all these cases, the corresponding quadratic equation will have no real roots.
e The solutions of a quadratic equation are represented by the x-intercept of the corresponding quadratic function. If our solutions are able to be represented as fraction, they will be rational numbers. Any graph with rational x-intercepts can represent this situation.
Notice that this is just an example graph, as there are infinitely many parabolas satisfying this condition. In all these cases the corresponding quadratic equation will have two roots, except for the case where the x-intercepts are the same. This is when the vertex lies on the x-axis. Then it would have just one root.
