How to find the equation of the line of symmetry

A parabola such as

#y = a_2x^2+a_1x+a_0#

can be put in the so called line of symmetry form by
choosing #c,x_0, y_0# such that

#y = a_2x^2+a_1x+a_0 equiv c(x-x_0)^2+y_0#

where #x = x_0# is the line of symmetry. Comparing coefficients we have

#{ (a_0 - c x_0^2 - y_0 = 0), (a_1 + 2 c x_0 = 0), (a_2 - c = 0) :}#

solving for #c, x_0, y_0#

# { (c = a_2), (x_0 = -a_1/(2 a_2)),( y_0 = (-a_1^2 + 4 a_0 a_2)/(4 a_2)) :} #

In the present case we have #c = -4, x_0 = 3/4, y_0 =-23/4# then

#x = 3/4# is the symmetry line and in symmetry form we have

#y = -4(x-3/4)^2-23/4#

A quadratic equation is a polynomial equation of degree 2 . The standard form of a quadratic equation is

0 = a x 2 + b x + c

where a , b and c are all real numbers and a ≠ 0 .

If we replace 0 with y , then we get a quadratic function

y = a x 2 + b x + c

whose graph will be a parabola .

The axis of symmetry of this parabola will be the line x = − b 2 a . The axis of symmetry passes through the vertex, and therefore the x -coordinate of the vertex is − b 2 a . Substitute x = − b 2 a in the equation to find the y -coordinate of the vertex. Substitute few more x -values in the equation to get the corresponding y -values and plot the points. Join them and extend the parabola.

Example 1:

Graph the parabola y = x 2 − 7 x + 2 .

Compare the equation with y = a x 2 + b x + c to find the values of a , b , and c .

Here, a = 1 , b = − 7 and c = 2 .

Use the values of the coefficients to write the equation of axis of symmetry .

The graph of a quadratic equation in the form y = a x 2 + b x + c has as its axis of symmetry the line x = − b 2 a . So, the equation of the axis of symmetry of the given parabola is x = − ( − 7 ) 2 ( 1 ) or x = 7 2 .

Substitute x = 7 2 in the equation to find the y -coordinate of the vertex.

y = ( 7 2 ) 2 − 7 ( 7 2 ) + 2 = 49 4 − 49 2 + 2 = 49 − 98 + 8 4 = − 41 4

Therefore, the coordinates of the vertex are ( 7 2 , − 41 4 ) .

Now, substitute a few more x -values in the equation to get the corresponding y -values.

x	y = x 2 − 7 x + 2
0	2
1	− 4
2	− 8
3	− 10
5	− 8
7	2

Plot the points and join them to get the parabola.

Example 2:

Graph the parabola y = − 2 x 2 + 5 x − 1 .

Compare the equation with y = a x 2 + b x + c to find the values of a , b , and c .

Here, a = − 2 , b = 5 and c = − 1 .

Use the values of the coefficients to write the equation of axis of symmetry.

The graph of a quadratic equation in the form y = a x 2 + b x + c has as its axis of symmetry the line x = − b 2 a . So, the equation of the axis of symmetry of the given parabola is x = − ( 5 ) 2 ( − 2 ) or x = 5 4 .

Substitute x = 5 4 in the equation to find the y -coordinate of the vertex.

y = − 2 ( 5 4 ) 2 + 5 ( 5 4 ) − 1 = − 50 16 + 25 4 − 1 = − 50 + 100 − 16 16 = 34 16 = 17 8

Therefore, the coordinates of the vertex are ( 5 4 , 17 8 ) .

Now, substitute a few more x -values in the equation to get the corresponding y -values.

x	y = − 2 x 2 + 5 x − 1
− 1	− 8
0	− 1
1	2
2	1
3	− 4

Plot the points and join them to get the parabola.

Example 3:

Graph the parabola x = y 2 + 4 y + 2 .

Here, x is a function of y . The parabola opens "sideways" and the axis of symmetry of the parabola is horizontal. The standard form of equation of a horizontal parabola is x = a y 2 + b y + c where a , b , and c are all real numbers and a ≠ 0 and the equation of the axis of symmetry is y = − b 2 a .

Compare the equation with x = a y 2 + b y + c to find the values of a , b , and c .

Here, a = 1 , b = 4 and c = 2 .

Use the values of the coefficients to write the equation of axis of symmetry.

The graph of a quadratic equation in the form x = a y 2 + b y + c has as its axis of symmetry the line y = − b 2 a . So, the equation of the axis of symmetry of the given parabola is y = − 4 2 ( 1 ) or y = − 2 .

Substitute y = − 2 in the equation to find the x -coordinate of the vertex.

x = ( − 2 ) 2 + 4 ( − 2 ) + 2 = 4 − 8 + 2 = − 2

Therefore, the coordinates of the vertex are ( − 2 , − 2 ) .

Now, substitute a few more y -values in the equation to get the corresponding x -values.

y	x = y 2 + 4 y + 2
− 5	7
− 4	2
− 3	− 1
− 1	− 1
0	2
1	7

Plot the points and join them to get the parabola.

How do you find the equation of the line of symmetry for a parabola?

The axis of symmetry always passes through the vertex of the parabola . The x -coordinate of the vertex is the equation of the axis of symmetry of the parabola. For a quadratic function in standard form, y=ax2+bx+c , the axis of symmetry is a vertical line x=−b2a .

How did you get the line of symmetry?

When you can divide a figure or an object into two even matching halves, you say that the figure has line symmetry. This figure can be divided in one way, vertically. If you tried to divide it horizontally, the two sides would not match. Therefore, you can say that the butterfly has bilateral symmetry.