3.9: Transformations of y = x²
Introduction
In this lesson you will learn that the graph of \begin{align*}y=x^2\end{align*} can undergo changes. These changes will include changing the vertex of (0, 0) and changing the \begin{align*}y-\end{align*}values of 1, 4 and 9. These changes will be used to draw the graph of \begin{align*}y=x^2\end{align*} as it undergoes the various changes.
Objectives
The lesson objectives for The Transformations of \begin{align*}y=x^2\end{align*} are:
- Understanding horizontal and vertical translations.
- Understanding a vertical stretch.
- Understanding a vertical reflection.
- The identity property of addition
- Applying these transformations to graph the parabola.
Transformations of y = x²
Introduction
In this concept you will learn to apply changes to the graph of \begin{align*}y=x^2\end{align*} to create a new parabola that no longer has its vertex at (0, 0) and no longer has \begin{align*}y-\end{align*}values of 1, 4 and 9. These changes are known as transformations and the transformations of \begin{align*}y=x^2\end{align*} can be readily detected on the graph.
Guidance
This is the graph of \begin{align*}y=x^2\end{align*}.
This is the graph of \begin{align*}y=x^2\end{align*} that has undergone transformations.
The vertex of the parabola on the right is (3, 1). The sides of the parabola open upward but they appear steeper and longer than those on the left. These transformations will be examined in the examples.
Example A
The blue parabola is the graph of \begin{align*}y=x^2\end{align*}. Its vertex is (0, 0). The red graph is the graph of \begin{align*}y=x^2\end{align*} that has been moved four units to the right. When the graph undergoes a slide of four units to the right, it has undergone a horizontal translation of +4. The vertex of the red graph is (4, 0). A horizontal translation changes the \begin{align*}x-\end{align*}coordinate of the vertex of the graph of \begin{align*}y=x^2\end{align*}.
Example B
The blue parabola is the graph of \begin{align*}y=x^2\end{align*}. Its vertex is (0, 0). The red graph is the graph of \begin{align*}y=x^2\end{align*} that has been moved four units to the right and three units upward. When the graph undergoes a slide of four units to the right, it has undergone a horizontal translation of +4. When the graph undergoes a slide of three units upward, it has undergone a vertical translation of +3.The vertex of the red graph is (4, 3). A horizontal translation changes the \begin{align*}x-\end{align*}coordinate of the vertex of the graph of \begin{align*}y=x^2\end{align*} while a vertical translation changes the \begin{align*}y-\end{align*}coordinate of the vertex.
Example C
This is the graph of \begin{align*}y=x^2\end{align*}. The points are plotted from the vertex as right and left one and up one-half, right and left 2 and up two, right and left three and up four and one-half. The original \begin{align*}y-\end{align*}values of 1, 4 and 9 have been divided by two or multiplied by one-half. When the \begin{align*}y-\end{align*}values are multiplied, the \begin{align*}y-\end{align*}values either increase or decrease. This transformation is known as a vertical stretch.
Example D
This is the graph of \begin{align*}y=x^2\end{align*}. The points are plotted from the vertex as right and left one and down one-half, right and left 2 and down two, right and left three and down four and one-half. The original \begin{align*}y-\end{align*}values of 1, 4 and 9 have been multiplied by one-half and then were made negative because the graph was opening downward. When the \begin{align*}y-\end{align*}values become negative, the direction of the opening is changed from upward to downward. This transformation is known as a vertical reflection. The graph is reflected across the \begin{align*}x-\end{align*}axis.
Vocabulary
- Horizontal translation
- The horizontal translation is the change in the base graph \begin{align*}y=x^2\end{align*} that shifts the graph right or left. It changes the \begin{align*}x-\end{align*}coordinate of the vertex.
- Transformation
- A transformation is any change in the base graph \begin{align*}y=x^2\end{align*}. The transformations that apply to the parabola are a horizontal translation \begin{align*}(HT)\end{align*}, a vertical translation \begin{align*}(VT)\end{align*}, a vertical stretch \begin{align*}(VS)\end{align*} and a vertical reflection \begin{align*}(VR)\end{align*}.
- Vertical Reflection
- The vertical reflection is the reflection of the image graph in the \begin{align*}x-\end{align*}axis. The graph opens downward and the \begin{align*}y-\end{align*}values are negative values.
- Vertical Stretch
- The vertical stretch is the change made to the base function \begin{align*}y=x^2\end{align*} by stretching (or compressing) the graph vertically. The vertical stretch will produce an image graph that appears narrower (or wider) then the original base graph of \begin{align*}y=x^2\end{align*}.
- Vertical Translation
- The vertical translation is the change in the base graph \begin{align*}y=x^2\end{align*} that shifts the graph up or down. It changes the \begin{align*}y-\end{align*}coordinate of the vertex.
Guided Practice
1. Use the following tables of values and identify the transformations of the base graph \begin{align*}y=x^2\end{align*}.
\begin{align*}& X \qquad -3 \qquad -2 \qquad -1 \qquad 0 \qquad 1 \qquad 2 \qquad 3\\ & Y \qquad \quad 9 \qquad \quad \ 4 \qquad \quad \ 1 \qquad \ 0 \qquad 1 \qquad 4 \qquad 9\end{align*}
\begin{align*}& X \qquad -4 \qquad -3 \qquad -2 \qquad -1 \qquad 0 \qquad 1 \qquad 2\\ & Y \qquad \ \ 15 \qquad \quad \ 5 \qquad -1 \qquad -3 \quad -1 \quad \ \ \ 5 \quad \ \ 15\end{align*}
2. Identify the transformations of the base graph \begin{align*}y=x^2\end{align*}.
3. Draw the image graph of \begin{align*}y=x^2\end{align*} that has undergone the following transformations:
\begin{align*}VR: \text{YES}\end{align*}
\begin{align*}VS: \frac{1}{2}\end{align*}
\begin{align*}VT: +2\end{align*}
\begin{align*}HT: \ -3\end{align*}
Answers
1. To identify the transformations from the tables of values, determine how the table of values for \begin{align*}y=x^2\end{align*} compare to the table of values for the new image graph.
- The \begin{align*}x-\end{align*}values have moved one place to the left. This means that the graph has undergone a horizontal translation of -1.
- The \begin{align*}y-\end{align*}coordinate of the vertex is -3. This means that the graph has undergone a vertical translation of -3. The vertex is easy to pick out from the tables since it is the point around which the corresponding points appear.
\begin{align*}& X \qquad -4 \qquad -3 \qquad -2 \qquad -1 \qquad 0 \qquad 1 \qquad 2\\ & Y \qquad \ \ 15 \qquad \quad \ 5 \qquad -1 \qquad -3 \quad -1 \quad \ \ \ 5 \quad \ \ 15\end{align*}
- The points from the vertex are plotted left and right one and up two, left and right two and up eight. This means that the base graph has undergone a vertical stretch of 2.
- The \begin{align*}y-\end{align*}values move upward so the parabola will open upward. Therefore the image is not a vertical reflection.
2. The vertex is (1, 6). The base graph has undergone a horizontal translation of +1 and a vertical translation of +6. The parabola opens downward, so the graph is a vertical reflection. The points have been plotted such that the \begin{align*}y-\end{align*}values of 1 and 4 are now 2 and 8. It is not unusual for a parabola to be plotted with five points rather than seven. The reason for this is the vertical stretch often multiplies the \begin{align*}y-\end{align*}values such that they are difficult to graph on a Cartesian grid. If all the points are to be plotted, a different scale must be used for the \begin{align*}y-\end{align*}axis.
3. The vertex is the values \begin{align*}(HT, VT)\end{align*} which is (-3, 2). The \begin{align*}y-\end{align*}values of 1, 4 and 9 must be multiplied by \begin{align*}\frac{1}{2}\end{align*} to create values of \begin{align*}\frac{1}{2}, 2\end{align*} and \begin{align*}4 \frac{1}{2}\end{align*}. The graph is a vertical reflection which means the graph opens downward and the \begin{align*}y-\end{align*}values become negative.
Summary
The base graph of the quadratic function of \begin{align*}y=x^2\end{align*} can undergo transformations that will change its appearance. The shape of the parabola which resembles a ‘U’ does not change but its location on the Cartesian grid can change. The vertex of (0, 0) will change if the image graph undergoes either a horizontal translation and/or a vertical translation. The transformations cause the parabola to slide left or right and up or down.
The points of the base graph are plotted from the vertex left and right one and up one, left and right two and up fore, left and right three and up nine. If the image graph undergoes a vertical stretch, these values of 1, 4 and 9 can increase if the stretch is a whole number. This will produce a parabola that will appear to be narrower than the original base graph. If the vertical stretch is a fraction, the values of 1, 4 and 9 will decrease. This will produce an image graph that will appear to be wider than the original base graph.
The final transformation that may occur is a vertical reflection. The graph is reflected in the \begin{align*}x-\end{align*}axis. The \begin{align*}y-\end{align*}values are all plotted downward as they will all be negative values. This causes the parabola to open downward.
Problem Set
The following table represents transformations to the base graph \begin{align*}y=x^2\end{align*}. Draw an image graph for each set of transformations.
Number | \begin{align*}VR\end{align*} | \begin{align*}VS\end{align*} | \begin{align*}VT\end{align*} | \begin{align*}HT\end{align*} |
---|---|---|---|---|
1. | NO | \begin{align*}3\end{align*} | \begin{align*}-4\end{align*} | \begin{align*}-8\end{align*} |
2. | YES | \begin{align*}2\end{align*} | \begin{align*}5\end{align*} | \begin{align*}6\end{align*} |
3. | YES | \begin{align*}\frac{1}{2}\end{align*} | \begin{align*}3\end{align*} | \begin{align*}-2\end{align*} |
4. | NO | \begin{align*}1\end{align*} | \begin{align*}-2\end{align*} | \begin{align*}4\end{align*} |
5. | NO | \begin{align*}\frac{1}{4}\end{align*} | \begin{align*}1\end{align*} | \begin{align*}-3\end{align*} |
For each of the following graphs, list the transformations of \begin{align*}y=x^2\end{align*}.
Answers
The following table represents...
Number | \begin{align*}VR\end{align*} | \begin{align*}VS\end{align*} | \begin{align*}VT\end{align*} | \begin{align*}HT\end{align*} |
---|---|---|---|---|
1. | NO | \begin{align*}3\end{align*} | \begin{align*}-4\end{align*} | \begin{align*}-8\end{align*} |
2. | YES | \begin{align*}2\end{align*} | \begin{align*}5\end{align*} | \begin{align*}6\end{align*} |
3. | YES | \begin{align*}\frac{1}{2}\end{align*} | \begin{align*}3\end{align*} | \begin{align*}-2\end{align*} |
4. | NO | \begin{align*}1\end{align*} | \begin{align*}-2\end{align*} | \begin{align*}4\end{align*} |
5. | NO | \begin{align*}\frac{1}{4}\end{align*} | \begin{align*}1\end{align*} | \begin{align*}-3\end{align*} |
For each of the following graphs...
- \begin{align*}VR: \text{YES}; \ VS: 4; \ VT: +7; \ HT: 0\end{align*}
- \begin{align*}VR: \text{NO}; \ VS: 2; \ VT: +4; \ HT: -5\end{align*}
- \begin{align*}VR: \text{YES}; \ VS: 3; \ VT: 0; \ HT: -3\end{align*}
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