The vertex of the graph of $y=(x-1)^2-5$ is ($\square$, $\square$).
Answer (1)
Recognize that the equation is in vertex form: y = a ( x − h ) 2 + k .
Identify the values of h and k from the given equation: h = 1 and k = − 5 .
State the vertex as ( h , k ) .
The vertex of the graph is ( 1 , − 5 ) .
Explanation
Understanding the Problem The given equation is y = ( x − 1 ) 2 − 5 . We need to find the vertex of the graph.
Identifying Vertex Form The equation is already in vertex form, which is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. In our case, a = 1 , h = 1 , and k = − 5 .
Finding the Vertex Therefore, the vertex of the parabola is ( 1 , − 5 ) .
Examples
Understanding the vertex of a parabola is useful in many real-world applications. For example, if you're launching a projectile, the vertex represents the maximum height the projectile will reach. Similarly, in business, if you have a cost function that is a parabola, the vertex can represent the point of minimum cost. Knowing how to find the vertex allows you to optimize these kinds of scenarios.
