The rules of transformations can be used to recognize the relationships between quadratic functions, even when not given information in the same format. Study the slides below to see how you can interpret transformations in a variety of ways.
Transformations Described in Words
The function \( g(x) \) is a parabola, opening upwards, with a vertex at \( (1,2) \).
The function \( g(x) \) described above is a transformation of the parent function \( f(x) = x^{2} \).
The function \( g(x) \) opens upwards, like the parent function, but the vertex has moved. An equation for \( g(x) \) could be \( g\left( x \right) = \left( x - 1 \right)^{2} + 2 \).
Transformations Described in a Table
\( \displaystyle x \)
\( \displaystyle g(x) \)
\( \displaystyle - 4 \)
\( \displaystyle - 16 \)
\( \displaystyle - 3 \)
\( \displaystyle - 9 \)
\( \displaystyle - 2 \)
\( \displaystyle - 4 \)
\( \displaystyle - 1 \)
\( \displaystyle - 1 \)
\( \displaystyle 0 \)
\( \displaystyle 0 \)
\( \displaystyle 1 \)
\( \displaystyle - 1 \)
\( \displaystyle 2 \)
\( \displaystyle - 4 \)
\( \displaystyle 3 \)
\( \displaystyle - 9 \)
\( \displaystyle 4 \)
\( \displaystyle - 16 \)
The function \( g\left( x \right) \), described above is a transformation of the parent function \( f(x) = x^{2} \).
The function \( g(x) \) closely resembles the parent function, but all of the \( y \)-values are negative. So, \( g\left( x \right) = - x^{2} \).
Transformations Described in Graphs
A parabola labeled \( g\left( x \right) \) that opens down, with a vertex at \( (0, - 1) \).
The function \( g\left( x \right) \) shows transformations of the parent function \( f(x) = x^{2} \).
The graph of \( g\left( x \right) \) shows two transformations:
The parent function has been flipped: \( f(x) = x^{2} \rightarrow - x^{2} \)
The vertex has been shifted down one unit: \( f(x) = x^{2} \rightarrow x^{2} - 1 \)
So, \( g\left( x \right) = - x^{2} - 1 \).
Transformations Described in Equations
\( \displaystyle g\left( x \right) = \left( x + 1 \right)^{2} \)
The function \( g(x) \) shows a transformation of the parent function \( f(x) = x^{2} \).
The equation of \( g\left( x \right) \) shows that \( x^{2} \) has been shifted one unit to the left: \( g\left( x \right) = \left( x - \left( - 1 \right) \right)^{2} \).
Let's Practice
Practice recognizing transformations. Look at each representation of a function and identify how the parent function \( f(x) = \left| x \right| \) has been transformed. When you think you know, click on the card to see the answer.
The function \( f(x) = \left| x \right| \) has been shifted down one unit. The translated function is \( g(x) = \left| x \right| - 1 \).
The function \( f(x) = \left| x \right| \) has been shifted three units to the right, and two units up. The translated function is \( g\left( x \right) = \left| x - 3 \right| + 2 \).
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