Look at the parabola below. How is this parabola different from \begin{align*}y=-x^2\end{align*}? What do you think the equation of this parabola is?
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Khan Academy Graphing a Quadratic Function
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 red parabola is (3, 1). The sides of the parabola open upward but they appear steeper and longer than those on the blue parabola.
As shown above, you can apply changes to the graph of \begin{align*}y=x^2\end{align*} to create a new parabola (still a 'U' shape) 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.
The vertex of (0, 0) will change if the parabola undergoes either a horizontal translation and/or a vertical translation. These transformations cause the parabola to slide left or right and up or down.
If the parabola undergoes a vertical stretch, the \begin{align*}y\end{align*}-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 less than 1, the values of 1, 4 and 9 will decrease. This will produce a parabola that will appear to be wider than the original base graph.
Finally, a parabola can undergo a vertical reflection that will cause it to open downwards as opposed to upwards. For example, \begin{align*}y=-x^2\end{align*} is a vertical reflection of \begin{align*}y=x^2\end{align*}.
Example A
Look at the two parabolas below. Describe the transformation from the blue parabola to the red parabola. What is the coordinate of the vertex of the red parabola?
Solution:
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
Look at the two parabolas below. Describe the transformation from the blue parabola to the red parabola. What is the coordinate of the vertex of the red parabola?
Solution:
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
Look at the parabola below. How is this parabola different from \begin{align*}y=x^2\end{align*}? What do you think the equation of this parabola is?
Solution:
This is the graph of \begin{align*}y=\frac{1}{2}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.
Concept Problem Revisited
This is the graph of \begin{align*}y=-\frac{1}{2}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, a vertical translation, a vertical stretch and a vertical reflection.
- 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 a vertical reflection, a vertical stretch by a factor of \begin{align*}\frac{1}{2}\end{align*}, a vertical translation up 2 units, and a horizontal translation left 3 units.
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.
- 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 given by the horizontal and vertical translations and 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.
Practice
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. VR = Vertical Reflection, VS = Vertical Stretch, VT = Vertical Translation, HT = Horizontal Translation.
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*} |
6. | YES | \begin{align*}1\end{align*} | \begin{align*}-4\end{align*} | \begin{align*}0\end{align*} |
7. | NO | \begin{align*}2\end{align*} | \begin{align*}3\end{align*} | \begin{align*}1\end{align*} |
8. | YES | \begin{align*}\frac{1}{8}\end{align*} | \begin{align*}0\end{align*} | \begin{align*}2\end{align*} |
For each of the following graphs, list the transformations of \begin{align*}y=x^2\end{align*}.
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