2.1 Quadratic Functions and Models
A quadratic function is a function of the form where a,b, and c are real numbers and .
- Called "parabolas" and have a line of symmetry (discussed later) and vertex
- if , the vertex is a minimum and the parabola opens up
- if , the vertex is a maximum and the parabola opens down
The standard form of a quadratic function is
- The graph of f is a parabola whose axis is the vertical line and whose vertex
is the point (h, k).
- If , the parabola opens upward and if , the parabola opens downward
Ex: Graph the function and find the vertex and axis of symmetry.
Factor out the 2 from the first two terms
Complete the square of
The vertex is (-2, -3) and the axis of symmetry is
Given , the vertex is located at with the axis of symmetry located
at .
Ex: Locate the vertex and axis of symmetry of without graphing.
Therefore the vertex is (1, 4) with the axis as
Ex: Graph by determining whether the graph opens up or down and by finding
its vertex, axis of symmetry, y-intercepts, and x-intercepts.
- Since , the parabola opens up
- so vertex is at (3, 0)
- The axis of symmetry is
- The y-intercept is (0, 9) since
- The x-intercept(s) are
so it is (3, 0)
Graph by hand on board
Ex: Write the equation of the parabola whose vertex is (1, 2) and that passes through the point (0, 0)
- Since the vertex is at (h, k), using standard form we have
- Passing through the point (0, 0) gives us (x, y) to substitute
So the equation is
HW: p. 134 1-8 all, 13-51 odd, 75-83 odd, 86, 92