how to graph parabolas?

Question

how to solve y=x2+2x-8

Answer 1

How do you graph y = x^2 + 2x – 8?

 

It's easy to graph y = x^2; just make a table of values, graph each point in the table, and draw a smooth curve between the points.

 

x | 0 | ±1 | ±2 | ± 3 | ...

y | 0 |  1  |  4   |  9   | ...

 

Notice that the left and right "sides" of the parabola are reflections of each other over the line x = 0, the y-axis, which is called the parabola's Axis of Symmetry. The Vertex is (0,0) and is the intersection point of the Axis of Symmetry and the function.

 

It's easy to translate the parabola by h in the x direction and k in the y direction:

 

y = x^2 → y–k = (x–h)^2, or y = (x–h)^2 + k

 

Every table point (x,y) → (x+h,y+k).

 

y = (x–h)^2 + k is called the Vertex Form because you can easily pick of the coordinates of the Vertex (h,k) and the Axis of Symmetry, x = h. This is the preferred form for graphing.

 

The problem's equation:

 

y = x^2 + 2x – 8 is in Standard Form.

 

Expanding the Vertex Form,

 

y = x^2 – 2hx + h^2 + k, and comparing to Standard Form,

y = x^2 + 2x – 8, we see that:

 

2 = –2h ==> h = –1, and

–8 = h^2 + k ==> k = –8 – (–1)^2 = –9.

 

So for this problem:

 

y = x^2 + 2x – 8 = (x+1)^2 – 9

 

To graph this parabola first draw dashed lines for x = –1 and y = –9. Then treat these dashed lines as a translated coordinate system and use the above table of values to graph points on it.

 

It's as if you drew the parabola on a transparent plastic sheet placed on top of a coordinate grid and then shifted the entire plastic sheet 1 unit left and 9 units down.

Answer 2

X^2 + 2X -8

 

  first a=1 , parabola open upward and vertex is minimum point

 

  Vertex ( -2/2, f( -1) ) = ( -1, -9)

 

 

    Axis of Symmetry : X = -1

 

    X intercepts:

                        X^2 + 2X - 8 = ( X +4 ) ( X -2) = 0   X = -4 , X =2

 

    Y intercept: ( 0 ,-8)

 

    Next : Graph.