Question

How to find the formula of a quadratic function knowing that it contains the points \[\left( { - 2,2} \right),\left( {0,1} \right),\left( {1, - 2.5} \right)\]?

Answer 1

Hint:
Here we will firstly write the general equation of the quadratic equation. Then we will put the values of the given points in the equation and solve it to get the value of \[a,b,c\] of the equation. Then we will put these values of \[a,b,c\] in the standard formula of the quadratic equation to get the final formula of the quadratic function which contains the given points.

Complete step by step solution:
Given points are \[\left( { - 2,2} \right),\left( {0,1} \right),\left( {1, - 2.5} \right)\].
We know that the genera form of the quadratic equation is given as
\[ \Rightarrow a{x^2} + bx + c = y\]
It is given that this quadratic function contains the given points. So this point will satisfy the equation. Therefore, we get
\[ \Rightarrow a{\left( { - 2} \right)^2} + b\left( { - 2} \right) + c = 2,a{\left( 0 \right)^2} + b\left( 0 \right) + c = 1,a{\left( 1 \right)^2} + b\left( 1 \right) + c = - 2.5\]
We will simply the equations we get
\[ \Rightarrow 4a - 2b + c = 2,c = 1,a + b + c = - 2.5\]
From these equations we get the value\[c = 1\]. So we will put the value of \[c\] in the other two equations, we get
\[ \Rightarrow 4a - 2b + 1 = 2,a + b + 1 = - 2.5\]
\[ \Rightarrow 4a - 2b = 1,a + b = - 3.5\]…………………………\[\left( 1 \right)\]
Now we will solve this equation to get the value of \[a,b\]. Therefore we will multiply the second equation by 2, we get
\[ \Rightarrow 4a - 2b = 1,2a + 2b = - 7\]
Now we will add both the equations to get the value of \[a\]. Therefore, we get
\[ \Rightarrow 4a - 2b + 2a + 2b = 1 - 7\]
\[ \Rightarrow 6a = - 6\]
\[ \Rightarrow a = - 1\]
Now we will put the value of \[a\] in any of the equation \[\left( 1 \right)\] to get the value of \[b\]. Therefore, we get
\[ \Rightarrow 4\left( { - 1} \right) - 2b = 1\]
\[ \Rightarrow - 4 - 2b = 1\]
\[ \Rightarrow 2b = - 4 - 1 = - 5\]
\[ \Rightarrow b = \dfrac{{ - 5}}{2} = - 2.5\]
Now we will put the values of \[a,b,c\] in the general form of the quadratic equation to get the actual quadratic equation, we get
\[ \Rightarrow - {x^2} - 2.5x + 1 = y\]

Hence the formula of a quadratic function knowing that it contains the points \[\left( { - 2,2} \right),\left( {0,1} \right),\left( {1, - 2.5} \right)\] is equal to \[y = - {x^2} - 2.5x + 1\].

Note:
Here the equation given in the question is the linear equation in which the highest exponent of the variable x is one. A quadratic equation is an equation in which the highest exponent of the variable x is two and a quadratic equation has only two roots. Roots are those values of the equation where the value of the equation becomes zero. For any equation numbers of roots are always equal to the value of the highest exponent of the variable x. Also, the cubic equations are the equation in which the highest exponent of the variable is 3. Also, we know that for an equation the number of its roots is equal to the value of the highest exponent of the variable of that equation which means for a cubic equation there are three roots of the equation.
Quadratic equation: \[a{x^2} + bx + c = 0\]
Cubic equation: \[a{x^3} + b{x^2} + cx + d = 0\]