Quadratic equations: Drawing parabolas
Transformations of parabolas
We have seen how to draw the parabola for a quadratic. Now we will learn how to transform the quadratic #y=x^2# and next easily draw the parabola of the new formula.
Transformations
We can transform the formula #y=x^2# in three different ways.
Transformations  Examples  
1 
We shift the graph of #y=x^2# up by #\green q#. The new formula is \[y=x^2+\green q\] 
geogebra plaatje

2 
We shift the graph of #y=\blue{x}^2# to the right by #\blue p#. The new formula is \[y=\left(x\blue p\right)^2\] 
geogebra plaatje

3 
We multiply the graph of #y=x^2# with #\purple a# relative to the #x#axis. The new formula is \[y=\purple a x^2\] 
geogebra plaatje

When we compare the formulas, we see that the new formula is obtained by adding #4# to the given formula. Therefore, the shape of the graph is the same as of the old formula, and the new graph is given by shifting the old one up #4#.
The intersection with the #y#axis of the old graph is #\rv{0,0}#. Because the function #4# shifts upward, the coordinates of the intersection with the #y#axis of the new formula are equal to: #\rv{0,4}#.
The vertex of the given formula is equal to #\rv{2,4}#. Because the new graph shifts up by #4#, the coordinates of the vertex of the new graph are equal to: #\rv{2,0}# .
The old graph intersects the #x#axis at the point with the #x# value #4#. In the new dashed graph, this point is moved #4# units. Therefore, the coordinates of the new point are: #\rv{4,4}#.
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