Recall the intercept form 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 intercept form of our function.
y=- 3(x-3)(x+5)
⇕
y= - 3(x- 3)(x-( - 5))
We can see that a= - 3, p= 3, and q= - 5. Therefore, the x-intercepts occur at ( 3,0) and ( - 5,0).
Find and Graph the Axis of Symmetry
The axis of symmetry is halfway between (p,0) and (q,0). Since we know that p=3 and q=- 5, the axis of symmetry of our parabola is halfway between (3,0) and (- 5,0).
x=p+q/2 ⇒ x=3+( - 5)/2=- 2/2=- 1
We found that the axis of symmetry is the vertical line x=- 1.
Find and Plot the Vertex
Since the vertex lies on the axis of symmetry, its x-coordinate is - 1. To find the y-coordinate, we will substitute - 1 for x in the given equation.
The y-coordinate of the vertex is 32. Therefore, the vertex is the point (- 1,32).
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 32. We can write the domain and range of the function using this information.
Domain:& all real numbers
Range:& y ≤ 32
