Every function, regardless of its degree, has a graph and each one is represented in a different way. The graph of a 1st degree function is a straight line that can be increasing or decreasing. The graph of a 2nd degree function will be either a downward or upward concavity parabola.
Every 2nd degree function is formed from the general form f (x) = ax2 + bx + c, with
a ≠ 0.
At first, to build a graph of any 2nd degree function, just assign values to x and find corresponding values for the function. Therefore, we will form ordered pairs, with them we will build the chart, see some examples:
Example 1:
Given the function f(x) = x2 – 1. This function can be written as follows: y = x2 – 1.
We will assign any value to x and substituting in the function we will find the value of y, forming ordered pairs.
y = (-3)2 – 1
y = 9 - 1
y = 8
(-3,8)
y = (-2)2 – 1
y = 4 - 1
y = 3
(-2,3)
y = (-1)2 – 1
y = 1 - 1
y = 0
(-1,0)
y = 02 – 1
y = -1
(0,-1)
y = 12 – 1
y = 1 - 1
y = 0
(1,0)
y = 22 – 1
y = 4 - 1
y = 3
(2,3)
y = 32 – 1
y = 9 - 1
y = 8
(3,8)
Distributing the ordered pairs in the Cartesian plane we will build the graph.
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The graph in this example has the concavity facing upwards, we can relate the concavity to the value of the coefficient a, when a > 0 the concavity will always be facing upwards.
Example 2:
Given the function f(x) = -x2. We will assign any value to x and substituting in the function we will find the value of y, forming ordered pairs.
y = -(-3)2
y = - 9
(-3,-9)
y = -(-2)2
y = - 4
(-2,-4)
y = -(-1)2
y = -1
(-1,-1)
y = -(0)2
y = 0
(0,0)
y = -(1)2
y = -1
(1,-1)
y = -(2)2
y = -4
(2,-4)
y = -(3)2
y = -9
(3,-9)
Distributing the ordered pairs in the Cartesian plane we will build the graph.
The graph in example 2 has the concavity facing downwards, as it was said in the conclusion of example 1 that the concavity is related to the value of the coefficient a, when a < 0 the concavity will always be turned to low.
by Danielle de Miranda
Graduated in Mathematics
Brazil School Team
Would you like to reference this text in a school or academic work? Look:
RIGONATTO, Marcelo. "Concavity of a Parable"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/concavidade-uma-parabola.htm. Accessed on June 28, 2021.
