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 x-axis. 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
• y-intercept
• x-intercepts
• domain
• range
• minimum/maximum value
Graph example to determine the characteristics of a function.
Characteristics of graph

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)

Characteristics of function: vertex
Characteristics of function: vertex

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

characteristics of function: axis of symmetry
Characteristics of function: axis of symmetry

Y-intercept

The y-intercept of a parabola is the value of y at the point where the parabola crosses the y axis.

In this question, the y-intercept = -8

characteristics of function: y-intercept
Characteristics of function: y-intercept

How to find x intercepts of a parabola

The x-intercept 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 x-intercepts = -2, 4

characteristics of function: x-intercept
Characteristics of function: x-intercept

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\inR

characteristics of function: domain
Characteristics of function: domain

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

characteristics of function: range
Characteristics of function: range

Minimum/maximum value

This parabola has a minimum value of -9

characteristics of function: maximum and minimum
Characteristics of function: maximum and minimum

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

Three properties that are universal to all quadratic functions: 1) The graph of a quadratic function is always a parabola that either opens upward or downward (end behavior); 2) The domain of a quadratic function is all real numbers; and 3) The vertex is the lowest point when the parabola opens upwards; while the vertex is the highest point when the parabola opens downward.

Lessons

  • 1.
    From the graph of the parabola, determine the:
    • vertex
    • axis of symmetry
    • y-intercept
    • x-intercepts
    • domain
    • range
    • minimum/maximum value
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Characteristics of quadratic functions

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