Comparison to the graph of f(x)=x^2: There is a vertical stretch by a factor of 2 and a horizontal translation left 3 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.
g(x)=2(x+3)^2 ⇔ g(x)= 2(x- ( - 3))^2
We can see that a= 2 and h= - 3. Therefore, the vertex is ( - 3,0), and the axis of symmetry is x= - 3. 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
2(x+3)^2
g(x)=2(x+3)^2
- 4
2( - 4+3)^2
2
- 5
2( - 5+3)^2
8
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.
Both graphs open up.
The graph of g(x)=2(x+3)^2 is narrower than the graph of f(x)=x^2.
The axis of symmetry of the parent function is the y-axis. The axis of symmetry of the given function is x=- 3.
The vertex of the given function, (- 3,0), is to the left of the vertex of the parent function, (0,0).
From the graph and the observations above, we can conclude that the graph of g is a vertical stretch by a factor of 2 and a horizontal translation left 3 units of the graph of f.
