Characteristics of quadratic functions
How to find the vertex of a parabola
When you graph a quadratic function, you’ll get a parabolic graph. By sketching it out, you can tell a lot from its sketch! One of the things you can determine is the vertex of its parabola.
The vertex is known as the turning point of the graph. It could be a maximum turning point or a minimum turning point depending on which way the graph’s opening is.
How to find axis of symmetry
To find the axis of symmetry, find the line that divides the parabola from the graphed quadratic function into exactly half. It’s also known as the vertex of the parabola, since it is at the point of the vertex where the graph is divided into half.
Domain and range of a parabola
What does the domain and range tell us about a graph? The domain is all the values of x that will give us real values of y in a function. The range is all the values of y that gives us real values of x. It helps tell us which values of x and y exist in a function. If you needed the real numbers definition, they are numbers that are rational.
For parabolic functions, the domain is usually from negative infinity to infinity. That’s because the graph opens up a certain direction and then continues on outwards along the negative and positive xaxis. This happens when a graph is concave up or concave down. Parabolic functions do, however, have a more defined range depending on where the graph is.
Practice problems
Let’s go through a question that will help you look at all the aspects of a quadratic graph.
Question:
From the graph of the parabola, determine the:
• vertex
• axis of symmetry
• yintercept
• xintercepts
• domain
• range
• minimum/maximum value
You can see video for this question from here:
https://www.youtube.com/watch?v=KRwb4YhQPwA
Solution:
Vertex:
The vertex of a parabola is the place where it turns; hence, it is also called the turning point. It is is the low or high point of the curve, sometimes called the maximum or minimum.
In this question, the vertex = (1, 9)
Axis of symmetry:
The graph of a quadratic function is a parabola. The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. The axis of symmetry always passes through the vertex of the parabola. The x coordinate of the vertex is the equation of the axis of symmetry of the parabola.
In this question, axis of symmetry is x = 1
Yintercept
The yintercept of a parabola is the value of y at the point where the parabola crosses the y axis.
In this question, the yintercept = 8
How to find x intercepts of a parabola
The xintercept of a parabola is the value of x at the point where the parabola crosses the x axis. A parabola can only have either none or two x intercepts
In this question, the xintercepts = 2, 4
Domain
The domain of a function is the set of all real values of x that will give real values for y. For quadratic functions, the domain is x = all real numbers. It can be presented as: x$\in$R
Range
The range of a function is the set of all real values of y that you can get by plugging real numbers into x. In this question, the range is y$\geq$9
Minimum/maximum value
This parabola has a minimum value of 9
If you wanted more practice problems, try out this quiz that has more practice problems dealing with quadratic functions.
For related concepts and lessons, check out even and odd functions, polynomial functions, and characteristics of polynomial graphs.
Characteristics of quadratic functions
Lessons

1.
From the graph of the parabola, determine the:
• vertex
• axis of symmetry
• yintercept
• xintercepts
• domain
• range
• minimum/maximum value