In this write-up we will explore the function
for different values of a.
Do you have any predictions about the shape of the graph as we change the value of a?
First, let's look at the case for a=0.
Now, let's vary the values for positive a values (0 to 5) and look what happens.
Here is a starting list of observations from the animation:
1) As the a values get close to zero the graph gets closer to a line.
2) As the a values increase, the parabola narrows in width towards the y=0 line, but never reaching it. Thus, there is an asymptote at y=0.
3) The parabolas are symmetric by the y=0 line.
4) The parabolas are concave up for a>0.
Next, let's vary the values for negative a-values (-5 to 0) and see what happens to the graph.
Here is a list of observations for the negative a animation:
1) The graph gets closer to a line as the a value approaches zero, which is similar to the positive a animation.
2) As the a values decrease, the parabola narrows in width towards the y=0 line, but never reaching it. Thus, there is an asymptote at y=0 , which is also true for positive a values.
3) The parabolas are symmetric by the y = 0 line, which is true for positive a values too.
4) The parabolas are concave down for a<0, which is opposite the positive values of a.
Do you see any additional observations?
What about a common point to all of the graphs we have explored, including
y = x + 2? Solution
Why would there be a common point to all of these graphs? Solution
Now, let's explore equations of the form,
First, we'll take a look at the animation of the b-value ranging from -5 to 5.
Notice that the vertice of the parabola is changing as the b-values are changed. For negative b-values, the vertice shifts to the right (increasing x-values) and down (decreasing the y-values). For positive b-values, the vertice shifts to the left (decreasing the x-values) and down (decreasing negative y-values).
Also, observe that the shape remains to be a parabola that is equal in width.
What conclusions can you draw from our observations about how the b-value affects the graph?
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