What does the graph of the function look like? Let's investigate this by starting out with a=1 and b=c=0. So the graph of F(x) is:
This is our standard parabola. The vertex is at (0,0) and it opens upward. So let's graph F(x) with several values for a and still b=c=0.
So the graphs for F(x) are:
Conjecture 1: Make a conjecture about the graph of as "a" changes. Include information about the vertex, direction that it opens, and relative widths of the graphs.
Email me your conjecture (be sure to label it as Conjecture 1): cmccord@coe.uga.edu
Perhaps we need to see more graphs of for different values of "a".
Does this change your conjecture about the graph of for different values of "a"?
If the answer is no, skip Conjecture 2. If the answer is yes, make a new conjecture.
Conjecture 2: Make a conjecture about the graph of as "a" changes. Include information about the vertex, direction that it opens, and relative widths of the graphs.
Email me your conjecture (be sure to label it as Conjecture 2): cmccord@coe.uga.edu
Put your conjectures to work.
Prediction 1:
Predict what the graphs of the following look like:
Include the following in your predictions:
direction the graphs open
the vertex
relative widths of the graphs
anything else you determine signifcant
Email your predictions to me: cmccord@coe.uga.edu
This first graph is with a=1, b=0, and c=1: (The graph of is included as a reference.)
Let's place several graphs together in order to form a conjecture:
O.K., let's vary "a" again but keep c=3 (still b=0):
Conjecture 3: Make a conjecture about the graph of as "a" and "c" change. Include information about the vertex, direction that it opens(i.e. concavity), and relative widths of the graphs.
Email me your conjecture (be sure to label it as Conjecture 3): cmccord@coe.uga.edu
Now let's investigate the graph of when a, b, and c are nonzero.
The vertex of is (-0.5,0.75) and the vertex of is (0.5,0.75). Write the vertices in fractional form (this may help you see a pattern).
Let's try several more graphs before making a conjecture:
The vertex of is (-1,0) and the y-intercept is (0,1).
The vertex of is (1,0) and the y-intercept is (0,1).
The vertex of is (1/3,-4/3); the y-intercept is (0,-1); and the x-intercepts are (1,0) and (-1/3,0).
The vertex of is (-1/3,-4/3); the y-intercept is (0,-1); and the x-intercepts are (-1,0) and (1/3,0).
It appears that the value of "b" effects the location of the vertex. Are you ready to make a conjecture?
Let's look at a few more examples before making a conjecture.
The vertex of is (2,-9); the y-intercept is (0,-5); and the x-intercepts are (-1,0) and (5,0).
The vertex of is (-2,-9); the y-intercept is (0,-5); and the x-intercepts are (-5,0) and (1,0).
The vertex of is (-1/2,-25/4); the y-intercept is (0,-6); and the x-intercepts are (-3,0) and (2,0).
The vertex of is (1/2,-25/4); the y-intercept is (0,-6); and the x-intercepts are (-2,0) and (3,0).
Have you developed a conjecture yet? Try graphing with a=c=1 and animating b (from -10 to 10). Then try this switching to a=-1 and c=1.
Click here to see this animation.
Conjecture 4: Make a conjecture about the graph of . Include information about the vertex, direction that it opens(i.e. concavity), relative widths of the graphs, y-intercept, and x-intercepts. Discuss what you think the affect of changing "b" is on the graph.
Email me your conjecture (be sure to label it as Conjecture 4): cmccord@coe.uga.edu
Questions:
Do you think that is the best form for the quadratic function in order to predict the graph of the function?
Why or why not?
What are the advantages and disadvantages of this form and the form ?
Which form of the quadratic function do you prefer to use?
Email your answers: cmccord@coe.uga.edu
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