can the vertex be the y-intercept

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 x2 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 = ax2 + 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) = 2x2 + 8x + 6 As you can see, the leading coefficient (of the x2 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: The quadratic function f(x) = 2x2 + 8x + 6 is concave up, since a = 2 is positive. Thus, the vertex is a minimum of the function. 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 = ax2 + 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) = -3x2 + 9x - 6 As you can see, the leading coefficient (of the x2 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: The quadratic function f(x) = -3x2 + 9x - 6 is concave down, since a = -3 is negative. Thus, the vertex is a maximum of the function. 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) = ax2 + bx + c or y = ax2 + 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 = ax2 + bx + c has one solution for y = 0. This occurs when the discriminant is zero, or when b2 – 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 b2 – 4ac = 0 (or b2 = 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 = 3x2 + 18x + 27 We have a = 3, b = 18, and c = 27, which means: b2 – 4ac=182 – 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. The quadratic function f(x) = 3x2 + 18x + 27 has its vertex on the x-axis, since the discriminant is zero. 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) = ax2 + bx + c or y = ax2 + bx + c where a is not zero. The y-intercept occurs where x = 0. This gives us y = ax2 + bx + cy = a(0)2 + b(0) + cy = 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 + ky = a(x – 0)2 + cy = ax2 + 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: b2 = 4acb = 0 Taking both equations together implies 02 = 4ac0 = 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 = ax2 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 = 2x2 + 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. The quadratic function f(x) = 2x2 + 8 has its vertex on the y-axis, since b = 0. 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) = ax2 + 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. 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