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.
g(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.
g(x)=-2 (x-4)(x+1)
⇕
g(x)= -2(x- 4)(x+( -1))
We can see that a= -2, p= 4, and q= -1. Therefore, the x-intercepts occur at ( 4,0) and ( -1,0).
Find and Graph the Axis of Symmetry
The axis of symmetry is halfway between (p,0) and (q,0). Since we know that p=4 and q=-1, the axis of symmetry of our parabola is halfway between (4,0) and (-1,0).
x=p+q/2 ⇒ x=4+( -1)/2=3/2
We found that the axis of symmetry is the vertical line x= 32.
Find and Plot the Vertex
Since the vertex lies on the axis of symmetry, its x-coordinate is 32. To find the y-coordinate, we will substitute 32 for x in the given equation.
The y-coordinate of the vertex is 252. Therefore, the vertex is the point ( 32, 252).
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 less than or equal to 252. We can write the domain and range of the function using this information.
Domain:& all real numbers
Range:& y ≤ 252
