1. Characteristics of Quadratics

2. Parabola
This graph is created by a 2nd degree equation, called a quadratic.
Standard form for a quadratic equation is f(x) = ax2 + bx + c.
The U-shape of the graph is called a parabola.
The graph continues both left and right, so the domain is all real numbers or R.
f(x) = x2 – 5x – 2

3. Coefficient of x2
The coefficient of x2 determines the direction in which the parabola opens.
For positive coefficients, the parabola opens up.
For negative coefficients,
the parabola opens down.
_ _
g(x) = – x2 + 8x – 12
+ +
f(x) = x2 + 8x + 13

4. Vertex
Every parabola has an extreme point where the curve turns back or changes direction. This point is called the vertex.
For parabolas that open up,
the vertex is a minimum, the
lowest y-value on the graph.
The range is y ≥ min.
For parabolas that open down,
the vertex is a maximum, the
highest y-value on the graph.
The range is y ≤ max.
maximum
f(x) = x2 + 8x + 13
g(x) = – x2 + 8x – 12
Range: y ≥ –3
Range: y ≤ 4
minimum

5. Axis of Symmetry
A parabola is symmetrical. The left half is the mirror image of the right half.
There is an imaginary vertical
line that runs through the
vertex and acts as a mirror for
the two halves of the parabola.
This line is called the axis of
symmetry.
The equation of the axis of
symmetry is always
x = x-coordinate of vertex.
vertex (2, 7)
axis of
symmetry
x = 2

6. Roots
The point at which a parabola crosses the x-axis is called an x-intercept, root, or zero (the value of y=0).
Parabolas can have zero roots, one root, or two roots, depending on how many times they cross the x-axis.
No roots
2 roots
1 root

7. y-intercepts
The point at which a parabola crosses the y-axis is called a y-intercept.
h(x) = – x2 + 5x – 3
The y-intercept may be found by looking at the constant in the standard form of the quadratic equation, f(x) = ax2 + bx + c.
The y-intercept is always at (0, c).
y-intercept
(0, –3)