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
Let's start by recalling 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 given function.
h(x)= - 4(x- 7)(x- 3)
We can see that a= - 4, p= 7, and q= 3. Therefore, the x-intercepts occur at ( 7,0) 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=7 and q=3, the axis of symmetry of our parabola is halfway between (7,0) and (3,0).
x=p+q/2 ⇒ x=7+ 3/2=10/2=5
We found that the axis of symmetry is the vertical line x=5.
Find and Plot the Vertex
Since the vertex lies on the axis of symmetry, its x-coordinate is 5. To find the y-coordinate, we will substitute 5 for x in the given equation.
The y-coordinate of the vertex is 16. Therefore, the vertex is the point (5,16).
Draw the Parabola
Finally, we will draw the parabola as a curve passing through the vertex and the x-intercepts.
We can see that there are no restrictions on the x-variable. Furthermore, h(x) — or y — takes values less than or equal to 16. We can write the domain and range of the function using this information.
Domain:& All real numbers
Range:& y ≤ 16
