In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-1) The graph passes through the point (-2,-3).

Verified step by step guidance

1. Recall that the vertex form of a parabola's equation is given by $y=a(x-h)2^{}+k$, where $(h,k)$ is the vertex.
2. Substitute the vertex coordinates $(-3,\; -1)$ into the vertex form equation, so it becomes $y\; =\; a(x\; +\; 3)^2\; -\; 1$.
3. Use the given point $(-2,\; -3)$ that lies on the parabola to find the value of $a$. Substitute $x\; =\; -2$ and $y\; =\; -3$ into the equation: $-3\; =\; a(-2\; +\; 3)^2\; -\; 1$.
4. Simplify the equation from step 3 to solve for $a$. Calculate the squared term and isolate $a$ on one side.
5. Write the final vertex form equation by substituting the found value of $a$ back into $y\; =\; a(x\; +\; 3)^2\; -\; 1$.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola's equation is y = a(x - h)^2 + k, where (h, k) is the vertex. This form makes it easy to identify the vertex and understand the parabola's shape and position on the coordinate plane.

Using a Point to Find 'a'

After writing the vertex form with the known vertex, substitute the coordinates of a given point on the parabola into the equation to solve for the coefficient 'a'. This determines the parabola's width and direction (upward or downward).

Graph Interpretation and Coordinate Substitution

Understanding how to interpret points on the graph and substitute their coordinates into an equation is essential. This process connects the geometric representation of the parabola with its algebraic equation.