Comparison to the graph of f(x)=x^2: There is a reflection in the x-axis and a horizontal translation left 2 units.
Practice makes perfect
To graph the parabola, we first need to identify the axis of symmetry and the vertex.
f(x)= a (x- h)^2
In this form, the axis of symmetry of the parabola is the vertical line x= h and the vertex is the point ( h,0). Now consider the given function.
h(x)=- (x-2)^2 ⇔ h(x)=( - 1)(x- 2)^2
We can see that a= - 1, h= 2. Therefore, the vertex is ( 2,0), and the axis of symmetry is x= 2. To graph the function we need to find two more points on the graph. Let's choose two x-values less than the x-coordinate of the vertex and make a table of values.
x
- (x-2)^2
h(x)=- (x-2)^2
0
-( 0-2)^2
- 4
1
-( 1-2)^2
- 1
Let's now plot the vertex and draw the axis of symmetry on a coordinate plane. We will also plot and reflect the obtained points across the axis of symmetry.
Let's draw a smooth curve that connects the five points. We will also draw the parent function f(x)=x^2.
From the graph above, we can note the following.
The graph of h(x)=- (x-2)^2 opens down, and the graph of f(x)=x^2 opens up.
The graphs do not have the same axis of symmetry. The axis of symmetry of the parent function is the y-axis. The axis of symmetry of the given function is x=2.
The vertex of the given function, (2,0), is to the right of the vertex of the parent function, (0,0).
We can conclude that the graph of h is a reflection in the x-axis and a horizontal translation right 2 units of the graph of f.
