Introduction to transformations
Before we begin the lesson, we must gain a better understanding of transformations.
A transformation what is added to a basic function in order to change its appearance or location on a graph. There are many ways a function can be translated including a translation, reflection, or a compression/stretch. A translation is a simple slide up, down, left, or right. A reflection is a flip over the x or y axis. A compression/stretch is moving all the x or y coordinates towards or away from the x or y axis. Are you confused yet? If so, have no fear! We'll go into detail more later.
We are almost ready to begin, but first we must establish a few necessary concepts that will apply to every transformation. These rules are:
When anything is grouped with x, or inside the parenthesis, the transformation will affect the x-axis. Also, any operation that you do is opposite when it is inside the parenthesis. For example, adding to a function would make it shift left and subtracting from a function would make it shift right. And, if x is multiplied by a whole number, it becomes a compression (5x becomes 1/5x) because the whole number becomes a fraction. It is the same for a fraction, 3/5x would become 5/3x when it is inside the parenthesis.
When something is not inside the parenthesis, the transformation affects the y-axis. If f(5x)+1 then the graph is moved up one unit from its original position. Vertical stretches and compressions happen outside the parenthesis as well, fractions create a compression and whole numbers create a stretch, which we will discuss later.
Here is a chart that explains the basic transformations. Take note of this, because it will be instrumental for our success today.
A transformation what is added to a basic function in order to change its appearance or location on a graph. There are many ways a function can be translated including a translation, reflection, or a compression/stretch. A translation is a simple slide up, down, left, or right. A reflection is a flip over the x or y axis. A compression/stretch is moving all the x or y coordinates towards or away from the x or y axis. Are you confused yet? If so, have no fear! We'll go into detail more later.
We are almost ready to begin, but first we must establish a few necessary concepts that will apply to every transformation. These rules are:
When anything is grouped with x, or inside the parenthesis, the transformation will affect the x-axis. Also, any operation that you do is opposite when it is inside the parenthesis. For example, adding to a function would make it shift left and subtracting from a function would make it shift right. And, if x is multiplied by a whole number, it becomes a compression (5x becomes 1/5x) because the whole number becomes a fraction. It is the same for a fraction, 3/5x would become 5/3x when it is inside the parenthesis.
When something is not inside the parenthesis, the transformation affects the y-axis. If f(5x)+1 then the graph is moved up one unit from its original position. Vertical stretches and compressions happen outside the parenthesis as well, fractions create a compression and whole numbers create a stretch, which we will discuss later.
Here is a chart that explains the basic transformations. Take note of this, because it will be instrumental for our success today.
Quadratic transformations
For each type of function we will discuss today, there is a parent function that corresponds with the specific type of function. For a quadratic, the parent function is f(x)=x^2, as shown below. To transform a quadratic function we evaluate the changes made in order to form a function that looks like this, a(bx-h)^2+k.
![Picture](/uploads/5/1/3/3/51337893/5748678.png?250)
The letter "a" represents a vertical stretch or compression (multiply this number to the y coordinates to apply). The letter "b" symbolizes a horizontal stretch or compression (multiply this number to the x coordinates to apply). A fraction symbolizes a compression and a whole number symbolizes a stretch. The letter "h" represents a horizontal shift (add this number to the x coordinates to apply). And, the letter "k" represents a vertical shift (add this number to the y coordinates to apply).
So if we asked you to transform x^2 into 2(x-3)^2+5 then you would make a table that shows the basic points for your parent function. The basic points for x^2 are (1,1), (0,0), and (-1,1). The transformations are a vertical stretch by a factor of 2, a horizontal shift right 3 units. Apply each transformation to every coordinate (the horizontal translations to the x coordinates and the vertical translations to the y coordinates) as shown below. Then, graph.
So if we asked you to transform x^2 into 2(x-3)^2+5 then you would make a table that shows the basic points for your parent function. The basic points for x^2 are (1,1), (0,0), and (-1,1). The transformations are a vertical stretch by a factor of 2, a horizontal shift right 3 units. Apply each transformation to every coordinate (the horizontal translations to the x coordinates and the vertical translations to the y coordinates) as shown below. Then, graph.
radical transformations
![Picture](/uploads/5/1/3/3/51337893/2617220_orig.png)
All of the same rules apply that we used for quadratic transformations, except, everything that happens revolves around the radicand. A radicand is similar to parenthesis. So, anything that happens under the radicand is the opposite of what the operation says and applies to the x axis. Anything that happens on the outside of the radicand applies to the y axis. The parent function of this type of function is f(x)=√(x). The equation we use to see the transformations is a√b(x-h) +k. The basic coordinates of this function are (0,0), (1,1), and (4,2).
So, if we asked you to translate √(x) into √1/4(x+2) -7 you would use the same strategy as above to translate the function. First, make a table of your basic points. Then, identify your transformations (in this case, a horizontal stretch by a factor of 4, a shift 2 to the left, and a shift 7 down. Apply the transformations to your table. Then, graph your points
So, if we asked you to translate √(x) into √1/4(x+2) -7 you would use the same strategy as above to translate the function. First, make a table of your basic points. Then, identify your transformations (in this case, a horizontal stretch by a factor of 4, a shift 2 to the left, and a shift 7 down. Apply the transformations to your table. Then, graph your points
logarithmic transformations
As with every transformation, the same rules apply to logarithmic transformations that we used for quadratic and radical transformations. Just like quadratic transformations, everything that is inside the parenthesis, are opposite. Inside the parenthesis, are the horizontal transformations. Everything outside the parenthesis are going to be a vertical transformation. However, everything that happens in a transformation, occurs around the logarithm. The parent function for a logarithmic transformation is log(x). Therefore, in order to transform it, everything would relate back to the parent function.
So, if we asked you to transform log(x) into 2log(1/3x-6)+4 you would first make a table of all the original points from the parent function. Then, you would describe your transformations, which is a vertical stretch by a factor of 2, and up 4 units. Then, there is a horizontal stretch by a factor of 3 and right 6 units. You want to make sure you remember PEMDAS. Apply to the table, and then graph.
So, if we asked you to transform log(x) into 2log(1/3x-6)+4 you would first make a table of all the original points from the parent function. Then, you would describe your transformations, which is a vertical stretch by a factor of 2, and up 4 units. Then, there is a horizontal stretch by a factor of 3 and right 6 units. You want to make sure you remember PEMDAS. Apply to the table, and then graph.