Draw the parabola 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 intercept form of a quadratic function.
f(x)=a(x-p)(x-q)
In this form, where a ≠0, the x-intercepts are p and q. Let's consider the intercept form of our function.
f(x)=(x+2)(x-3)
⇕
f(x)= 1(x-( - 2))(x- 3)
We can see that a= 1, p= - 2, and q= 3. Therefore, the x-intercepts occur at ( - 2,0) and ( 3,0).
Find and Graph the Axis of Symmetry
The axis of symmetry is halfway between (p,0) and (q,0). Since we know that p=- 2 and q=3, the axis of symmetry of our parabola is halfway between (- 2,0) and (3,0).
x=p+q/2 ⇒ x=-2+ 3/2=1/2
We found that the axis of symmetry is the vertical line x= 12.
Find and Plot the Vertex
Since the vertex lies on the axis of symmetry, its x-coordinate is 12. To find the y-coordinate, we will substitute 12 for x in the given equation.
The y-coordinate of the vertex is - 254. Therefore, the vertex is the point ( 12,- 254).
Draw the Parabola
Finally, we will draw the parabola through the vertex and the x-intercepts.
We can see above that there are no restrictions on the x-variable. Furthermore, the y-variable takes values greater than or equal to - 254. We can write the domain and range of the function using this information.
Domain:& all real numbers
Range:& y ≥ - 254
