In this lesson, our instructor Vincent Selhorst-Jones goes over the Properties of Quadratic Functions in detail. His lesson begins with parabolas and then goes into the axis of symmetry and vertex. Youll also learn about transformations, switching forms by completing the square, and how to find the vertex of a parabola when it is positive or negative. Vincent also goes over an alternative form to quadratics and how to analyze different parabolas. Before the lesson is over, youll have the chance to practice with four fully worked-out examples.
The graph of a quadratic function gives a parabola: a symmetric, cup-shaped figure.
Since the parabola is symmetric, we can draw a line that the parabola is symmetric around. This is called the axis of symmetry.
The point where the axis of symmetry crosses the graph is the vertex. Informally, we can think of this point as where the parabola "turns"-the place where the graph changes directions.
While parabolas can be very different from one another, they are still fundamentally similar. Any parabola can be turned into another parabola through transformations (see the lesson on Transformations of Functions).
With this realization in mind, we can convert any quadratic into the form
a ·(x−h)2 + k.
This form shows all the transformations that have been applied to the fundamental square function of x2.
We can convert into this format by completing the square (see the previous lesson, Completing the Square and the Quadratic Formula, for more information).
In this new form, it's easy to find the vertex. From transformations, we see the the vertex has been shifted horizontally by h and vertically by k. So, in this form, our vertex is at (h, k).
If our quadratic is in the standard form of f(x) = ax2+bx+c, the vertex is at
By knowing the vertex of a parabola, we can find the minimum or maximum that the quadratic attains.
Once we know the location of the vertex, it's extremely easy to find the axis of symmetry. The axis of symmetry runs through the vertex, so if the vertex is at (h,k), then the axis of symmetry is the vertical line x=h.
Properties of Quadratic Functions
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.