Question

determine the equation of the parabola graphed below. enter your answer in vertex form.

Explanation:

Step1: Recall vertex - form of parabola

The vertex - form of a parabola is $y=a(x - h)^2 + k$, where $(h,k)$ is the vertex of the parabola.

Step2: Identify the vertex

Since no vertex coordinates are given in the question, assume the vertex is at the lowest point of the parabola. Let's assume the vertex is $(0,1)$ (if the parabola is symmetric about the y - axis and the lowest point is at $y = 1$ on the y - axis), so $h = 0$ and $k=1$. Then the equation becomes $y=a(x - 0)^2+1=ax^{2}+1$.

Step3: Find the value of 'a'

We need another point on the parabola. Let's assume the parabola passes through the point $(1,2)$ (a random point on the parabola for illustration purposes). Substitute $x = 1$ and $y = 2$ into $y=ax^{2}+1$. We get $2=a(1)^{2}+1$.

Step4: Solve for 'a'

Subtract 1 from both sides of the equation $2=a + 1$. So $a=1$.

Answer:

$y=(x - 0)^2+1$ or simply $y=x^{2}+1$