Which variable is raised to the second power? Does it mean that the axis of symmetry of the parabola is a horizontal or a vertical line?
Vertex: (- 3, - 5) Focus: ( - 3, - 19/4 ) Directrix: y=- 21/4 Axis of Symmetry: x=- 3 Transformation: Vertical stretch by a factor of 4, a translation 3 units left, and a translation 5 units down.
Practice makes perfect
Before we begin, note that in the given equation the variable that is raised to the second power is x.
y=(x+3)^()darkorange2-5
Therefore, the axis of symmetry of the parabola is a vertical line.
Finding the Desired Information
Let's recall the general form of the equation for this type of parabola.
y=1/4 p(x- h)^2+ k
In order to match this form perfectly, let's rewrite our equation so that the parenthetical factor has a coefficient.
y=(x+3)^2-5 ⇔ y=1(x+3)^2-5
We need to identify the values of h, k, and p. Let's start with p. To do so, we will solve the equation 14 p=1. We set it equal to 1, because 1 is the coefficient of the parenthetical factor.
Knowing that p= 14, we can rewrite the equation.
y=1(x+3)^2-5
⇕
y=1/4( 14 ) (x-( - 3))^2+( - 5)
Now we have that h= - 3, k= - 5, and p= 14. By recalling the corresponding formulas we can find the vertex, focus, directrix, and axis of symmetry of the parabola.
Vertex
Focus
Directrix
Axis of Symmetry
Formula
( h, k)
( h, k+ p)
y= k- p
x= h
Value
( - 3, - 5)
( - 3, - 5+ 1/4) ⇓ ( - 3,- 19/4 )
y= - 5- 1/4 ⇓ y=- 21/4
x= - 3
Drawing the Parabola
Now, let's draw the parabola using the obtained information.
We can compare this graph to the standard graph with p=1 and vertex (0,0), y= 14x^2.
We see that the given function is a vertical stretch by a factor of 4, followed by a translation 3 units left and 5 units down.
