Question

what is the equation of the graph below?
(graph of a parabola opening upwards, vertex at (-2, 3))
options:
y = - (x - 2)² + 3
y = (x + 2)² + 3
y = - (x + 3)² + 2
y = (x - 3)² + 2

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. If \(a>0\), the parabola opens upward; if \(a<0\), it opens downward.

Step2: Identify the vertex from the graph

From the graph, the vertex (the lowest point of the upward - opening parabola) is at \((- 2,3)\). So \(h=-2\) and \(k = 3\). Also, since the parabola opens upward, \(a>0\).

Step3: Substitute \(h\), \(k\) and \(a\) into vertex form

Substituting \(h=-2\), \(k = 3\) and \(a = 1\) (since it opens upward and the coefficient of the squared term is positive in the options we are considering) into \(y=a(x - h)^2+k\), we get \(y=(x-(-2))^2 + 3=(x + 2)^2+3\). We can also check the other options:

• For \(y=-(x - 2)^2+3\), \(a=- 1<0\) (opens downward) and vertex \((2,3)\), which does not match the graph.
• For \(y=-(x + 3)^2+2\), \(a=-1<0\) (opens downward) and vertex \((-3,2)\), which does not match the graph.
• For \(y=(x - 3)^2+2\), vertex \((3,2)\), which does not match the graph.

Answer:

\(y=(x + 2)^2+3\)