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.