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Boundless Algebra
Quadratic Functions and Factoring
Graphs of Quadratic Functions
Parts of a Parabola
The graph of a quadratic function is a parabola, and its parts provide valuable information about the function.
LEARNING OBJECTIVES
Describe the parts and features of parabolas
KEY TAKEAWAYS
Key Points
The graph of a quadratic function is a U-shaped curve called a parabola.
The sign on the coefficient aa of the quadratic function affects whether the graph opens up or down. If a<0a<0, the graph makes a frown (opens down) and if a>0a>0 then the graph makes a smile (opens up).
The extreme point ( maximum or minimum ) of a parabola is called the vertex, and the axis of symmetry is a vertical line that passes through the vertex.
The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function.
Key Terms
vertex: The point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function.
axis of symmetry: A vertical line drawn through the vertex of a parabola around which the parabola is symmetric.
zeros: In a given function, the values of xx at which y=0y=0, also called roots.
Recall that a quadratic function has the form
f(x)=ax2+bx+cf(x)=ax2+bx+c.
where aa, bb, and cc are constants, and a≠0a≠0.
The graph of a quadratic function is a U-shaped curve called a parabola. This shape is shown below.

Parabola : The graph of a quadratic function is a parabola.
In graphs of quadratic functions, the sign on the coefficient aa affects whether the graph opens up or down. If a<0a<0, the graph makes a frown (opens down) and if a>0a>0 then the graph makes a smile (opens up). This is shown below.

Direction of Parabolas: The sign on the coefficient aa determines the direction of the parabola.
Features of Parabolas
Parabolas have several recognizable features that characterize their shape and placement on the Cartesian plane.
Vertex
One important feature of the parabola is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph.
Axis of Symmetry
Parabolas also have an axis of symmetry, which is parallel to the y-axis. The axis of symmetry is a vertical line drawn through the vertex.
yy-intercept
The y-intercept is the point at which the parabola crosses the y-axis. There cannot be more than one such point, for the graph of a quadratic function. If there were, the curve would not be a function, as there would be two yy values for one xx value, at zero.
xx-intercepts
The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of xx at which y=0y=0. There may be zero, one, or two xx-intercepts. The number of xx-intercepts varies depending upon the location of the graph (see the diagram below).

Possible xx-intercepts: A parabola can have no x-intercepts, one x-intercept, or two x-intercepts
Recall that if the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the function. These are the same roots that are observable as the xx-intercepts of the parabola.
Notice that, for parabolas with two xx-intercepts, the vertex always falls between the roots. Due to the fact that parabolas are symmetric, the
d, or Pi times diameter