To draw the graph of the given function, we will follow four steps.
- Identify and plot the .
- Find and graph the .
- Find and plot the .
- Draw the through the vertex and the points where the x-intercepts occur.
Let's go through these steps one at a time.
Identify and Plot the x-intercepts
Recall the of a quadratic function.
y=a(x−p)(x−q)
In this form, where a = 0, the x-intercepts are p and q. Let's consider the factored form of our function.
f(x)=-3(x+1)(x+7)⇕f(x)=-3(x−(-1))(x−(-7)) We can see that a=-3, p=-1, and q=-7. Therefore, the x-intercepts occur at (-1,0) and (-7,0).
Find and Graph the Axis of Symmetry
The axis of symmetry is halfway between (p,0) and (q,0). Since we know that p=-1 and q=-7, the axis of symmetry of our parabola is halfway between (-1,0) and (-7,0).
x=2p+q⇒x=2-1+(-7)=2-8=-4
We found that the axis of symmetry is the x=-4.
Find and Plot the Vertex
Since the vertex lies on the axis of symmetry, its
x-coordinate is
-4. To find the
y-coordinate, we will substitute
-4 for
x in the given equation.
f(x)=-3(x+1)(x+7) f(-4)=-3(-4+1)(-4+7) f(-4)=-3(-3)(3) f(-4)=9(3) f(-4)=27
The
y-coordinate of the vertex is
27. Therefore, the vertex is the point
(-4,27).
Draw the Parabola
Finally, we will draw the parabola through the vertex and the x-intercepts.