How do you find the vertex for #y=x^2+6x+7#?
2 Answers
Use the 'complete the square form' i.e.
Explanation:
Okay, so:
in order to get to the complete the square form, you need to halve the 'b' coefficient of your
right?
Let's expand the bracket, shall we?
Now, you'll notice that the 'c' isn't 7, but 9. That's where the 'n' comes in. The 'n' is supposed to be a simple calculation that gets your 'c' right; if your 'n' becomes a little bit too complicated, double check if you've done the halving/ expanding correctly!
We can get the 'c' to 7 by subtracting 2. The -2 becomes 'n'. So we get:
To get:
So now that you have your complete the square form done, think of what number gets ONLY the bracket to equal to 0. In this case, -3. Thus, -3 becomes your x value for the vertex.
However, you have the remaining 'n' part to think of as well, but since the bracket is already equal to 0,
your 'n' value becomes the y value for the vertex! So the vertex of the graph is:
(-3, -2)
Hope this helps!
An alternative to finding the vertex. See explanation for steps.
Vertex has a coordinate of
Explanation:
Graphing
In this case, the graph is curving up due to the fact that
Since it is a graph curving up, we know that the gradient at points on the graph would be increasing from a negative number when the value of x increases.
At the vertex, or in other words, the minimum point (in this case since it curves up) has a gradient of 0.
Why?
Because the minimum point is a point where the gradient changes from negative to positive.
Firstly, to find its vertex, we need to form a gradient function,
Differentiate
We know that at vertex, gradient or
We have the x-coordinate of the vertex. To find the y-coordinate, simply plug in
The vertex has a coordinate of
