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Suggested Problems from Text:
p. 251 #1-8, 10, 11, 15, 16, 18, 19, 21, 23, 24, 30, 33, 37, 38, 75
Graphs
Standard Form
Applications
Graphs
A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.
The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape. The picture below shows three graphs, and they are all parabolas.
All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point called the vertex of the parabola.
You know that two points determine a line. This means that if you are given any two points in the plane, then there is one and only one line that contains both points. A similar statement can be made about points and quadratic functions.
Given three points in the plane that have different first coordinates and do not lie on a line, there is exactly one quadratic function f whose graph contains all three points. The applet below illustrates this fact. The graph contains three points and a parabola that goes through all three. The corresponding function is shown in the text box below the graph. If you drag any of the points, then the function and parabola are updated.
Many quadratic functions can be graphed easily by hand using the techniques of stretching/shrinking and shifting (translation) the parabola y = x2 . (See the section on manipulating graphs.)
Example 1.
Sketch the graph of y = x2/2. Starting with the graph of y = x2, we shrink by a factor of one half. This means that for each point on the graph of y = x2, we draw a new point that is one half of the way from the x-axis to that point.
