Translating Parabolas

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Using Vertex Form

To translate the graph of a quadratic function, you can use the vertex form of a quadratic function,

y = a(x – h)2 + k

Properties of Graphs in Vertex Form

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When h is positive the graph shifts right; when h is negative the graph shifts left.

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When k is positive the graph shifts up; when k is negative the graph shifts down.

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The vertex is (h, k) and the axis of symmetry is the line x = h.

 

Example 1:
Name the vertex for the functions given below:
y = (x – 2)2 + 3

(2, 3)
y = 3(x + 1)2 + 5

(– 1, 5)
y = – 2(x +3)2

(– 3, 0)
y = (x + 4)2 – 5

(– 4, – 5)
y = x2 + 3

(0, 3)

Using Vertex Form to Graph a Parabola

 

Example 2: Graph y = (x – 2)2 – 1
  1. Graph the vertex (2, – 1)
  2. Find another point.  When x = 1
    y     = (1 – 2)2 – 1 = 0 Graph (1, 0)
  3. Graph the point corresponding to (1, 0).  It is 2 units to the right of the axis of symmetry at (3, 0).
  4. Sketch the curve.

 

Writing the Equation of a Parabola

Example 3:
Write the equation of the parabola shown below:
Notice that the vertex of the parabola is located at (3, 1).  Also note that the point (2, 3) is on the graph.
Use vertex form

y = a(x – h)2 + k
Substitute h = 3 and k = 1

y = a(x – 3)2 + 1
Substitute x = 2 and y = 3

3 = a(2 – 3)2 + 1
Simplify

3 = a + 1
Solve for a

a = 2
The equation of the parabola is

y = 2(x – 3)2 + 1

Converting to Vertex Form

Example 4:  Write y = 3x2 – 6x + 7
Find the x-coordinate of the vertex

Find the y-coordinate of the vertex.

y = 3(1)2 – 6(1) + 7

y= 4
The vertex is at (1, 4)
Write the vertex form.

y = 3(x – 1)2 + 4