Concavity of a parable

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.

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

Source: Brazil School - https://brasilescola.uol.com.br/matematica/concavidade-uma-parabola.htm