Topics
Discussion
Do you notice any visual similarities between the graphs below?
Solution:
Let's compare the three graphs by examining their symmetry and key points:
For the parabola y=x2:
The vertex is at (0,0).
For any x=a, the point is (a,a2), and for x=−a, the point is (−a,a2).
This means the graph is symmetric about the y-axis: every point on the right has a matching point at the same height on the left.
For the V-shaped graph y=∣x∣:
The corner is at (0,0).
For x=a, the point is (a,a); for x=−a, the point is (−a,a).
Again, the graph is symmetric about the y-axis, with the left and right rays mirroring each other.
For the cosine curve y=cosx:
The maximum is at (0,1).
The zeros are at (2π,0) and (−2π,0).
The minimums are at (π,−1) and (−π,−1).
The curve to the right of the y-axis is a mirror image of the curve to the left.
In all three cases, the graph for x>0 is a mirror image of the graph for x<0. Therefore, all three graphs are symmetric about the y-axis.
Discussion
Do you notice any visual similarities between the graphs below?
Solution:
Let's compare the three graphs by examining their symmetry and key points:
For the parabola y=x2:
The vertex is at (0,0).
For any x=a, the point is (a,a2), and for x=−a, the point is (−a,a2).
This means the graph is symmetric about the y-axis: every point on the right has a matching point at the same height on the left.
For the V-shaped graph y=∣x∣:
The corner is at (0,0).
For x=a, the point is (a,a); for x=−a, the point is (−a,a).
Again, the graph is symmetric about the y-axis, with the left and right rays mirroring each other.
For the cosine curve y=cosx:
The maximum is at (0,1).
The zeros are at (2π,0) and (−2π,0).
The minimums are at (π,−1) and (−π,−1).
The curve to the right of the y-axis is a mirror image of the curve to the left.
In all three cases, the graph for x>0 is a mirror image of the graph for x<0. Therefore, all three graphs are symmetric about the y-axis.