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Quadratic Functions

Quadratic functions (parabolas) have the following form:



 

The lowest or highest point on a quadratic graph is called the vertex. The x–coordinate of the vertex occurs at . This vertical line also forms the axis of symmetry of the graph, which means that if the graph were folded along its axis, the left and right sides of the graph would coincide.

In graphs of the form  if a > 0, then the graph opens up.

 

 
If a < 0, then the graph opens down.
 


 

By completing the square, the form  can be written as . You are not expected to know this form on the test. But it is a convenient form since the vertex occurs at the point (h, k) and the axis of symmetry is the line x = h.

We have been analyzing quadratic functions that are vertically symmetric. Though not as common, quadratic functions can also be horizontally symmetric. They have the following form:



 

The furthest point to the left on this graph is called the vertex. The y-coordinate of the vertex occurs at . This horizontal line also forms the axis of symmetry of the graph, which means that if the graph were folded along its axis, the top and bottom parts of the graph would coincide.

In graphs of the form  if a > 0, then the graph opens to the right and if a < 0 then the graph opens to the left.
 
Example

The graph of  and the graph of the line k intersect at (0, p) and (1, q). Which one of the following is the smallest possible slope of line k ?

Let’s make a rough sketch of the graphs.
 
Expressing  in standard form yields .
 
Since a = –1, b = 0, and c = 2, the graph opens to the left and its vertex is at (2, 0).
 
 
Since p and q can be positive or negative, there are four possible positions for line k (the y-coordinates in the graphs below can be calculated by plugging x = 0 and x =1 into the function ):
 
   
 
Since the line in the first graph has the steepest negative slope, it is the smallest possible slope.
Calculating the slope yields
 
 
The answer is (A).
 





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