Properties
What is a Quadratic Function?
A Quadratic Function is an equation where each x-value corresponds to only one y-value. The graph will appear as a "U" shaped graph. The graph is called a parabola. The parabolas below are examples of quadratic functions.
Hint: Remember that function notation is f(x) and the graph of an equation us y =. These two are interchangable and you will see both during this unit.
Key Properties of Graphs
The graph of a quadratic function may appear simple, but there are many different properties to the graph. The following is a list of the properties which you will study during this lesson:
- Positive or Negative Parabola
- Vertex of the Parabola
- Axis of Symmetry
- Maximum and Minimum Values
- Intervals of Increasing and Decreasing
- x-intercept(s)
- y-intercept
Positive or Negative Parabola
The standard form of a parabola, or quadratic equation, is:
y = ax2 + bx + c
If the value of a is greater than zero, the parabola will open up and will be a positive parabola.
If the value of a is less than zero, the parabola will down and will be a negative parabola.
The Vertex of a Quadratic
The graph below is a positive quadratic function.
The next property to identify is the vertex of the function. The vertex can be thought of as the turning point of the graph.
The Axis of Symmetry of a Quadratic
The graph below is a positive quadratic function.
The axis of symmetry is the vertical line that passes through the vertex. If you were to draw this graph on a piece of paper and fold along the axis of symmetry, each side of the graph would align.
Maximum and Minimum Values
The graph below is a positive quadratic function.
The minimum value of a quadratic is the lowest point on the graph. The minimum value is the same coordinate as the vertex of the function. You will have a minimum value if your graph opens up, or in other words, is positive.
The graph below is a negative quadratic function.
The maximum value of a quadratic is the highest point on the graph. The maximum value is the same coordinate as the vertex of the function. You will have a maximum value if your graph opens down, or in other words, is negative.
Increasing Interval of a Quadratic
The graph below is a positive quadratic function.
The interval where the graph is increasing occurs where the graph appears to be climbing uphill.
Decreasing Interval of a Quadratic
The graph below is a positive quadratic function.
The interval where the graph is decreasing occurs where the graph appears to be falling downhill.
The x-intercepts
The graph below is a positive quadratic function.
The x-intercepts of the graph are the intersections of the graph and the x-axis. The x-intercepts play an important role in the solution to the equation of the quadratic. Quadratic functions will have zero, one, or two x-intercepts.
Decreasing Interval of a Quadratic
The graph below is a positive quadratic function.
The y-intercept of the graph is the intersection of the graph and the y-axis. The y-intercept will also be the constant term of the function. Quadratic functions will have one y-intercept.
