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
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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
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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
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When we compare the formulas, we see that the new formula is obtained by adding #2# 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 #2#.
The intersection with the #y#-axis of the old graph is #\rv{0,-8}#. Because the function #2# shifts upward, the coordinates of the intersection with the #y#-axis of the new formula are equal to: #\rv{0,2}#.
The vertex of the given formula is equal to #\rv{1,1}#. Because the new graph shifts up by #2#, the coordinates of the vertex of the new graph are equal to: #\rv{1,3}# .
The old graph intersects the #x#-axis at the point with the #x# value #2#. In the new dashed graph, this point is moved #2# units. Therefore, the coordinates of the new point are: #\rv{2,2}#.
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