draw the graph of the polynomial p(x)=x^2-6x+9 find its zeroes
Answers
The first task here is to find the roots of the equation, that is where the graph intercepts the
x
axis. To do this, set
y
=
0
and then factorise:
x
2
+
6
x
+
9
=
0
(
x
+
3
)
2
=
0
→
x
=
−
3
In this case as
x
=
−
3
twice then the graph will "graze" x-axis at
x
=
−
3
, (intercept it only once). Note, if there were 2 distinct roots then the graph would cut straight the through the
x
axis at the 2 points and if there were no real roots then the graph would not intercept the
x
axis at all.
Find where the graph intercepts the
y
axis by setting
x
=
0
→
y
=
(
0
)
2
+
6
(
0
)
+
9
=
9
So our
y
intercept is at
(
0
,
9
)
We also have to find the turning point. This can be done in a number of ways, finding the midway point between the roots, setting the derivative equal to 0 or compete the square.
In this case we know that since
x
=
−
3
is the only root then that must also be the turning point.
As it is a quadratic where the
x
2
coefficient is positive then we know the graph will be a parabola with a minimum.
We now have all the information we need to graph the function. Simply mark on your points and sketch:
graph{x^2+6x+9 [-10, 10, -2, 10]}.Ans.