Brush up on your Algebra For all of the tutorials on nonlinear functions and models, you should be familiar with the algebra of exponents and radicals.
The relationship between two quantities is often best modeled by a curved line rather than a straight line. The simplest function with a graph that is not straight line is a quadratic function.
Quadratic Function
A quadratic function of the variable x is a function that can be written in the form
f(x) = ax2 + bx + c
Function form
y = ax2 + bx + c
Equation form
where a, b, and c are fixed numbers (with a ≠ 0).
Examples
Every quadratic function f(x) = ax2 + bx + c has a parabola as its graph. Whether it opens up ("concave up") or opens down ("concave down") depends on the sign of a . Press the "a > 0" and "a < 0" buttons to see the difference in the figure below.
Features of a Parabola
Features of a Parabola
The graph of f(x) = ax2 + bx + c (a ≠ 0) is a parabola with the following features:
Concavity If a > 0, the parabola is concave up; if a < 0 it is concave down.
Example The graph of f(x) = -3x2 - 6x - 3 has a = -3 < 0, so the graph is concave down (press the "a < 0" button above to see its shape.
Vertex The x coordinate of the vertex is -b/2a. The y coordinate is f(-b/2a).
Example The graph of f(x) = -3x2 - 6x - 3
has vertex with the following coordinates:
x coordinate =
-b
2a
=
-(-6)
2(-3)
= -1
y coordinate =
f
-b
2a
= f(-1) =
-3(-1)2 - 6(-1) - 3 = 0
y-Intercept The y-intercept is given by y = c.
Example The graph of f(x) = -3x2 - 6x - 3 has c = -3, so the y-intercept is given by y = -3.
x-Intercepts The x-intercepts, if they exist, are given by setting
ax2 + bx + c = 0
ans solving for x. To do this, you may beed to factor the quadratic (better if it works!) or use the quadratic formula:
x =
- b
b2 - 4ac
2a
For a quick review, see Solving Polynomial Equations in the on-line algebra review. Note: If the quadratic has no real factors (equivalently, if b2 - 4ac is negative) then there are no x-intercepts -- the parabola is entirely above or below the x-axis.
Example
The graph of f(x) = -3x2 - 6x - 3 has x-intercept obtained by solving the quadratic
-3x2 - 6x - 3 = 0
To solve, first divide both sides by -3 and then factor:
-3x2 - 6x - 3 = 0
x2 + 2x + 1 = 0
(x + 1)(x + 1) = 0
The only solution is x = 1, so this gives the only x-intercept.
y = -3x2 - 6x - 3
Let f(x) = 4x2 - 8x - 21.
1. Compute the coordinates of the vertex of its graph and click on its location in the following grid.
2. Now compute the y-intercept.
y-intercept
=
3. Now Compute the x-intercepts, and click on both of them in the grid below. (Try to be as accurate as possible.)
Let f(x) = -x2 + x - 3.
Now select (click on) the correct graph from the following (the gridlines are one unit apart):
The population of Roman Catholic nuns in the US during the last 25 years of the 90's can be modeled by