The vertex of a parabola is an extreme value of the function. However, it helps to know when it is a maximum or minimum and when it might touch a coordinate axis.

So, what do you need to know about the vertex of a parabola? **A parabola’s vertex is the minimum value when a > 0, and in this case, the parabola is convex (concave up) and opens upward (like a cup shape). A parabola’s vertex is the maximum value when a < 0, and in this case, the parabola is concave (concave down) and opens downward (like a dome shape).**

Of course, the vertex of a parabola can like on the x-axis, on the y-axis, and in some cases on both at the same time (that is, the vertex is at the origin).

In this article, we’ll answer some common questions about the vertex of a parabola and give some examples to make the concepts clear.

Let’s get started.

## Questions About The Vertex Of A Parabola

You might be familiar with what a parabola looks like, but you may still need to know when the vertex is a maximum or minimum and when it touches the x or y axes.

Let’s start with when a vertex is a maximum or minimum

### Is The Vertex A Minimum Or Maximum?

The vertex of a parabola can be a maximum or minimum depending on the sign of the leading (quadratic) coefficient, which tells us the shape of the graph of a quadratic.

Here is how to tell if the vertex is a maximum or minimum of a parabola:

**Look at the leading coefficient**(that is, the coefficient of the x^{2}term in the quadratic equation that the parabola comes from).**Look at the concavity of the parabola**(that is, examine the sign of the second derivative, f’’(x), of the quadratic function, f(x), that the parabola comes from).**Look at the graph of the parabola**(which way does it open: up or down?)

Let’s take a look at when each case occurs.

#### When Is The Vertex Of A Parabola The Minimum Point?

The vertex of a parabola is the minimum point when:

**The leading coefficient is positive**(that is, in the quadratic equation y = ax^{2}+ bx + c, we have a > 0).**The parabola is convex, or concave up**(that is, f’’(x) > 0, where f’’(x) is the second derivative of the quadratic function f(x) that the parabola comes from).**The parabola opens upward**(that is, the graph of the parabola is shaped like a cup, not a dome – it would hold water).

##### Example: The Vertex Of A Parabola Is The Minimum Point

Consider the quadratic function

**f(x) = 2x**^{2}+ 8x + 6

As you can see, the leading coefficient (of the x^{2} term) is a = 2. Since 2 is positive (a > 0), we know that the vertex of the corresponding parabola is a minimum value.

We can also find the second derivative of the function:

**f’(x) = 4x + 8**[first derivative of f(x)]**f’’(x) = 4**[second derivative of f(x)]

Since 4 is positive (f’’(x) > 0), we know that the function is convex (concave up). This confirms that the vertex of the parabola is a minimum.

Finally, we can also visually confirm that the vertex is a minimum by examining the graph of the parabola below:

#### When Is The Vertex Of A Parabola The Maximum Point?

The vertex of a parabola is the maximum point when:

**The leading coefficient is negative**(that is, in the quadratic equation y = ax^{2}+ bx + c, we have a < 0).**The parabola is concave, or concave down**(that is, f’’(x) < 0, where f’’(x) is the second derivative of the quadratic function f(x) that the parabola comes from).**The parabola opens downward**(that is, the graph of the parabola is shaped like a dome, not a cup – water would run off the top).

##### Example: The Vertex Of A Parabola Is The Maximum Point

Consider the quadratic function

**f(x) = -3x**^{2}+ 9x – 6

As you can see, the leading coefficient (of the x^{2} term) is a = -3. Since -3 is negative (a < 0), we know that the vertex of the corresponding parabola is a maximum value.

We can also find the second derivative of the function:

**f’(x) = -6x + 9**[first derivative of f(x)]**f’’(x) = -6**[second derivative of f(x)]

Since -6 is negative (f’’(x) < 0), we know that the function is concave (concave down). This confirms that the vertex of the parabola is a maximum.

Finally, we can also visually confirm that the vertex is a maximum by examining the graph of the parabola below:

### Can The Vertex Be The X-Intercept?

The vertex can be the x-intercept of a parabola in some cases (this happens when the vertex rests on the x-axis). We can find out when this will occur by combining what we know about the x-intercept, vertex, and solutions of a quadratic function.

First, remember that the standard form of a quadratic function is given by:

**f(x) = ax**^{2}+ bx + c

or

**y = ax**^{2}+ bx + c

where a is not zero.

Also, remember that a vertex is unique – a parabola has only one vertex. In this case, the vertex has a y-coordinate of zero, since it rests on the x-axis.

So, the corresponding quadratic equation y = ax^{2} + bx + c has one solution for y = 0. This occurs when the discriminant is zero, or when b^{2} – 4ac = 0.

(You can learn more about when a quadratic has exactly one real solution in my article here).

So, if the quadratic equation has b^{2} – 4ac = 0 (or b^{2} = 4ac), the corresponding parabola has its vertex on the x-axis (that is, the vertex is also the x-intercept).

#### Example: The Vertex Is The X-Intercept

Consider the quadratic function

**y = 3x**^{2}+ 18x + 27

We have a = 3, b = 18, and c = 27, which means:

**b**^{2}– 4ac**=18**^{2}– 4(3)(27)**=324 – 324****=0**

Since the discriminant is zero, the corresponding parabola will have its vertex on the x-axis, as you can see in the graph below.

### Can The Vertex & Y-Intercept Be The Same?

The vertex can be the y-intercept of a parabola in some cases (this happens when the vertex rests on the y-axis). We can find out when this will occur by combining what we know about the x-intercept, vertex, and solutions of a quadratic function.

First, remember that the standard form of a quadratic function is given by:

**f(x) = ax**^{2}+ bx + c

or

**y = ax**^{2}+ bx + c

where a is not zero.

The y-intercept occurs where x = 0. This gives us

**y = ax**^{2}+ bx + c**y = a(0)**^{2}+ b(0) + c**y = c**

In this case, the y-intercept is given by the point (x, y) = (0, c).

Recall that the vertex form of a parabola is given by

**y = a(x – h)**^{2}+ k

where a is as above and (h, k) is the vertex.

If we plug in (0, c) = (h, k) for the vertex (so h = 0 and k = c), we get:

**y = a(x – h)**^{2}+ k**y = a(x – 0)**^{2}+ c**y = ax**^{2}+ c

This suggests that the quadratic function must have a linear coefficient of zero, or b = 0.

So, if the quadratic equation has b = 0, then the corresponding parabola has its vertex on the y-axis (that is, the vertex is also the y-intercept).

Note that if we want the vertex to be at the origin (the intersection of the x-axis and y-axis), we need both:

**b**^{2}= 4ac**b = 0**

Taking both equations together implies

**0**^{2}= 4ac**0 = 4ac**

Since a is not zero, this implies c = 0. So, a quadratic must have b = 0 and c = 0 to have its vertex at the origin.

That is, only a quadratic of the form y = ax^{2} can have its vertex at the origin (a can be any nonzero value).

#### Example: The Vertex Is The Y-Intercept

Consider the quadratic function

**y = 2x**^{2}+ 8

We have a = 2, b = 0, and c = 8. Since b = 0, the corresponding parabola will have its vertex on the y-axis, as you can see in the graph below.

### Is The Vertex Of A Parabola Differentiable?

The vertex of a parabola is differentiable. Since a parabola comes from a quadratic equation (a polynomial of degree 2), we can differentiate as many times as we want and still get a differentiable function.

Note that for any parabola, the corresponding quadratic function has the form:

**f(x) = ax**^{2}+ bx + c

The derivative of this function is given by:

**f’(x) = 2ax + b**

The vertex of a parabola is at the point when x = -b / 2a, which gives us:

**f’(-b / 2a)****=2a(-b / 2a) + b****=-b + b****=0**

Meaning the derivative of the quadratic function is zero (the tangent line is flat, with a slope of zero) at the vertex.

## Conclusion

Now you know the answers to a few important questions about the vertex of a parabola. You also have some examples to help illustrate some of these ideas.

You can learn about how to change the shape of a parabola in my article here and the axis of symmetry of a parabola in my article here.

You can learn about the focus of a parabola and what it means here.

You can learn how to find the domain and range of a parabola here.

I hope you found this article helpful. If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

~Jonathon