Title
Properties of Parabolas


Problem Statement
Compare the graphs of quadratic functions in the form:  y = ax2 + bx + c

 

a.       What are the coordinates of the y-intercept?

b.      What must be true in order for the graph to reflect over the y-axis?

c.       What must be true in order for the graph to open down?

d.      What must be true in order for the graph to open up?

 

Problem setup

What happens to the parabola represented by the quadratic equation y = ax

 2  + bx + c  when the variables are changed from positive to negative? 

 

 

Plans to Solve/Investigate the Problem

We will use the graphing calculator to observe the quadratic function y = ax2 + bx + c to find the coordinates of the y-intercept.  We will try different values for a, b, and c in order to explore the results on the graph.

 

 

 

Investigation/Exploration of the Problem

a.)  In the function y=ax2 + bx + c, c is our y intercept and our constant.  When we first enter the function, we find that a, b, and c are all one and the parabola opens upward with the vertex at the point (-0.5, 0.75).  When we change the value of  a to 3 and keep b and c equal to one, the parabola moves to the right and becomes more narrow with the vertex at (-0.234375, 0.93041992).

 

Blue  y = ax2 + bx + c

Green y = 3x2 + bx + c

 

 

b.)      In order to make the parabola reflect over the y axis, we found that b must be 0.  At the point (0,0) the parabola is symmetrical.  In other words, all parabolas in the form y = ax2 + bx + c, where b is 0, because b x 0 is 0, leaves the equation y=ax2 + c, and will be reflected symmetrically over the y axis.

 

 

 

c. and d.)  When the value of a is a positive number in the equation y = x2 + x + 1,  so that y  = ax2 + bx + c, the parabola opens upward.  When the value of a is changed to a negative number, the parabola reverses and opens downward.  In our example, we  show the vertex moves to (0.25, 1.125) when a is -2. 

 

 

Purple

Red

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In our original investigations, we tried different values for a, b, and c.  When we returned a to one, and changed b to 4, with c remaining at one.  The parabola we found had x and y intercepts were both one.  Changing b to -4 shifted the parabola’s vertex to the first quadrant and kept the y intercept at one.  We wondered if changing b to -4 would move the parabola’s vertex further to the right.  In fact, the vertex became (2,-3) and the parabola increased in size.  Reversing the value of b to +4, the parabola moved to the mirror image position with the vertex at (-2,-3).

 

Red 

Purple

 

 

 

 

 

 

In an attempt to create another parabola that opens downward, we changed c to a negative number, but found that the parabola remained open upwards, with only the y intercept changing to -2.

Purple

Red

Blue

 

 

 

Extensions of the Problem

If ax2 + bx + c = 0, where is the solution on the graph?

 

 

The above graph is the graph 2x2 + x - 3 = y.   As we can see, the graph intersects the x axis at x= -1.5 and x = 1 which are the roots of the equation 2x2 + x – 3 = 0.   Actually this graph represents the roots of the equation, and  the roots of the equation ax2 + bx + c = 0 are where the graph intersects the x axis.

Author & Contact
Jill Jackson and Shirley Crawford.
 

Link(s) to resources, references, lesson plans, and/or other materials

 

www.mste.uiuc.edu/dildine/sketches/parabola.htm