To analyze the given quadratic function [tex]\( y = 5(x + 1)^2 - 3 \)[/tex], let's follow these steps:
1. Rewrite the Function in Vertex Form:
Notice that the function is already in the vertex form [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\( a = 5 \)[/tex], [tex]\( h = -1 \)[/tex], and [tex]\( k = -3 \)[/tex]. This form is useful for finding the vertex of the parabola.
2. Identify the Vertex:
The vertex form of the quadratic function provides the vertex directly as [tex]\((h, k)\)[/tex]. Here,
[tex]\[
h = -1 \,\text{and}\, k = -3.
\][/tex]
Therefore, the vertex of the parabola is [tex]\((-1, -3)\)[/tex].
3. Understand the Vertex:
- The vertex [tex]\((h, k)\)[/tex] represents the highest or lowest point on the graph of the quadratic function, depending on the sign of [tex]\(a\)[/tex].
- Since [tex]\( a = 5 \)[/tex] is positive, the parabola opens upwards, and the vertex represents the minimum point on the graph.
4. Confirm the Expression and Vertex Coordinates:
- The function we started with is [tex]\( y = 5(x + 1)^2 - 3 \)[/tex].
- The vertex of this function is found at the coordinates [tex]\((-1, -3)\)[/tex].
By performing these steps, we've found that the quadratic function [tex]\( y = 5(x + 1)^2 - 3 \)[/tex] has its vertex at the point [tex]\((-1, -3)\)[/tex]. The vertex form clearly indicates this result and confirms that the parabola opens upwards because the coefficient of the squared term ([tex]\(a\)[/tex]) is positive.
Therefore, the function and its vertex are:
[tex]\[
y = 5(x + 1)^2 - 3 \quad \text{with a vertex at} \quad (-1, -3).
\][/tex]
