Question Video: Given a Graph of a Quadratic Function Identify the Coordinates of the Vertex Mathematics • Third Year of Preparatory School

Video Transcript

Identify the coordinates of the vertex of the quadratic function 𝑓 of 𝑥 equals 𝑥 squared minus one.

We’re given the quadratic function 𝑓 of 𝑥 equals 𝑥 squared minus one and the graph of that function. First, we should recall what the vertex of a quadratic function is. The vertex is either a minimum or a maximum, sometimes called the turning point. Since this parabola opens upward, the vertex will be a minimum. By inspection, it looks like we can see the minimum here at an 𝑥-coordinate of zero and a 𝑦-coordinate of negative one, halfway between zero and negative two.

Sometimes we’re not able to identify the vertex of a function from its graph with accuracy, so it’s helpful to know another method for finding this. For a given function in vertex form 𝑦 equals 𝑎 times 𝑥 minus ℎ squared plus 𝑘. The vertex of the graph is located at ℎ, 𝑘. We can rearrange the function we were given so that it’s in vertex form. 𝑓 of 𝑥 equals one times 𝑥 minus zero squared plus negative one. This means that 𝑎 equals one, ℎ equals zero, and 𝑘 equals negative one. Since the vertex is the point ℎ, 𝑘, that would be equal to zero, negative one.

This matches what we see on the graph that the vertex of the function 𝑓 of 𝑥 equals 𝑥 squared minus one is located at the point zero, negative one.