Transformations on the Basic Parabola
Before you begin this lesson, you should already be familiar with the graph of y = x². In this lesson you will examine the standard form of a quadratic equation including how each value in the equation affects the graph. Your booklet covers this topics piece by piece over several pages - this page can be used as an overview of all of the different transformations.
The following is a quadratic equation in standard form :
In this equation, each variable other than x and y represent a different graphical transformation. They are:
a represents a vertical stretch or compression
h represents a horizontal shift (or slide)
k represents a vertical shift (or slide)
Each transformation starts from the graph of y = x². If you do not know how to create the graph of y = x², go find out before you continue here. All of these transformations are described below, and we will tie them all together at the end.
Vertical Stretch or Compression (a)
We will begin our look at transformations with the vertical stretch.
To perform a vertical stretch on a graph, multiply ALL y-coordinates by the stretch factor (a).
Let's look at an example:
Vertical Stretch Example 1
Sketch the graph of y = 3x².
Solution
As with all quadratic transformations, start with the graph of y = x². Three important points are labelled ((0,0), (-1,1) and (1,1)) Remember these points, they will be used for all of our transformation work.
The y-coordinates have been highlighted in red as these are what we work with when performing a vertical stretch. Remember the rule for performing a stretch:
To perform a vertical stretch on a graph, multiply ALL y-coordinates by the stretch factor (a).
The stretch factor (the a-value) in this question is 3. Multiply all of the values in red above by 3 to get the new y-coordinates for the new graph. (-1, 1) becomes the point (-1, 3). (0, 0) becomes (0, 0) (since 3 times 0 is 0). (1, 1) becomes the point (1, 3). The new (and final) graph is shown below:
Notice that the graph appears to have been 'stretched' upward. The larger the value of a, the larger the stretch. A negative 'a' value works the same way, but it flips the graph vertically over the x-axis. You will see this in the next example:
Vertical Stretch Example 2
Sketch the graph of y = -2x².
Solution
As with all of our transformations, start with the graph of y = x². (Again, the three important points are marked):
As with the last graph, multiply all of the y-coordinates of the 'important points' by the stretch factor (which in this case is -2). This gives us the following three new points: (-1, -2), (0, 0) and (1, -2). The new graph is shown below:
The graph has been 'flipped' as well as stretched. Some teachers teach the flip as a separate transformation. Just treat the flip however you learned it in class.
Purplemath.com has a nice example of how the graph of y = x² changes as the value of a changes. Just click on the link at the beginning of this sentence - the animation is at the beginning of the page that opens.
The h value in the general equation stands for the horizontal shift. This transformation means that you shift EVERY point to the left or right h units. Remember the following:
- If h is positive, you will shift the graph to the right.
- If h is negative, you will shift the graph to the left.
This can get a little tricky, as the form of the general equation throws a wrench into things. Look at the equation again:
Specifically, look at the part in the brackets: (x - h). The minus sign in the brackets makes things look 'backwards' when you are talking about the h value.
For example, (x - 3) has an h value of positive 3. How does that work? Look inside the brackets again (x - h). h HAS to be positive to keep that negative around. Substitute +3 in for h and you get: (x - (+3)) which turns into (x - 3).
On the flip side, (x + 4) has an h value of negative 4. Again, look at the general form in the brackets (x - h). h has to be negative to change the negative from the general equation to a positive. If you substitute -4 in for h you get (x - (-4)) which simplifies to (x + 4).
Got that straight? Let's look at an example:
Horizontal Shift Example
Sketch the graph of y = (x - 4)²
Solution
In this equation, you are given a value for h. In this case h is positive 4 (+4). (If you don't understand why, see above.) This means you must shift EVERY point 4 units to the right. Below is an animation of this transformation, starting from the graph of y = x² and finishing at the graph of y = (x-4)².
The last transformation we need to look at is a vertical shift. The vertical shift is represented by the variable k in the standard form of a quadratic equation. The following rules apply:
- If the value of k is positive, you shift EVERY point on the graph UP k units.
- If the value of k is negative, you shift EVERY point on the graph DOWN k units.
This transformation is very similar to a horizontal shift, except you are moving up and down and the signs aren't 'wonky'.
Vertical Shift Example
Sketch the graph of y = x² - 6
Solution
As you can see from the equation, the value of k is -6. That means that you need to shift every point on the graph of y = x² down 6 units. Below is an animation showing this transformation.
Putting It All Together
Of course, you usually won't be dealing with just one type of transformation at a time. You will usually need to complete multiple transformations to arrive at a final graph. The example below shows you a graph where you need to apply all three of the transformations that we looked at above.
Example
Sketch the graph of y = -3(x + 2)² + 4
Solution
When you need to perform multiple transformations, you must do them in order. Specifically, you do each transformation as you see it in the equation. a (the vertical stretch) comes first, then h (the horizontal shift) and lastly k (the vertical shift). Let's list what we have to do to the graph of y = x² in order:
- First we need to do the vertical stretch. Multiply all y - coordinates by a (which in this case is -3).
- Perform a horizontal shift of -2. Move all points 2 units to the left.
- Perform a vertical shift of +4. Move all points up 4 units.
The animation below shows each of these transformations in order and the final graph. Make sure you watch the animation from the beginning.
As you can see, this page does not look at this topic to the extent that your teacher does. You should make sure that you have ALL of the notes for this very important section, and be sure that you have completed all questions assigned in your booklet.
You may now try a couple of sample questions, or you can head back to the Quadratic Functions main page.