Question

challenge #1
plot a parabola through the points.
(hint: try typing ( y = x ^ { 2 } ))

Explanation:

Step1: Understand the general form of a parabola

The general form of a parabola is \(y = ax^{2}+bx + c\). Since the parabola passes through \((0,0)\), substituting \(x = 0\) and \(y=0\) into \(y=ax^{2}+bx + c\) gives \(0=a\times0^{2}+b\times0 + c\), so \(c = 0\). The equation becomes \(y=ax^{2}+bx\).

Step2: Use the other two points

Let's assume the two non - origin points. Suppose one point is \((1,1)\) and \((- 1,1)\) (based on the symmetry of the parabola \(y=x^{2}\)). Substitute \(x = 1,y = 1\) into \(y=ax^{2}+bx\): \(1=a\times1^{2}+b\times1=a + b\). Substitute \(x=-1,y = 1\) into \(y=ax^{2}+bx\): \(1=a\times(-1)^{2}+b\times(-1)=a - b\).

Step3: Solve the system of equations

We have the system \(

$$\begin{cases}a + b=1\\a - b=1\end{cases}$$

\). Add the two equations: \((a + b)+(a - b)=1 + 1\), \(2a=2\), so \(a = 1\). Substitute \(a = 1\) into \(a + b=1\), we get \(1 + b=1\), so \(b=0\).

Answer:

The equation of the parabola is \(y=x^{2}\)