a The x-intercepts represent the real zeroes of the function.
B
b The x-intercepts represent the real zeroes of the function.
C
c The x-intercepts represent the real zeroes of the function.
A
aExample Graph:
B
bExample Graph:
C
cExample Graph:
Practice makes perfect
a The x-intercepts represent the real zeroes of the function, so we should have 3. It also has 2 imaginary zeroes. The least degree it can have is 3+2 = 5. Furthermore, a degree 5 polynomial function has at most 5-1=4 turning points. We are ready to sketch our graph — it will have 4 turning points and intersect the x-axis 3 times.
Notice that this is just an example, as there are infinitely many graphs satisfying the given conditions.
b Recall that the x-intercepts represent the real zeroes of the function. We can sketch our function by making sure that it has exactly 4 x-intercepts.
Notice that this is just an example, as there are infinitely many graphs satisfying the given conditions.
c We need a polynomial with 2 imaginary zeroes. The lowest possible degree for it is 2. Since the x-intercepts represent the real zeroes of the function, we can use a quadratic function with no x-intercepts. According to the Fundamental Theorem of Algebra, it must have 2 imaginary zeroes if it has no real ones.
Notice that this is just an example, as there are infinitely many graphs satisfying the given conditions.
