Stretch vs Shrink (Compression)
Stretch vs Shrink (Compression) - How it Works - Video
Horizontal and Vertical Shifts
Horizontal and Vertical Shifts:
Vertical Shifts
On the top left we have a vertical shift up.
On the top the right we have a vertical shift down.
For vertical shifts, the y-values are change and the x-values are the same.
Horizontal Shifts
On the bottom left we have a horizontal shift left.
On the bottom right we have a horizontal shift right.
For horizontal shifts, the x-values are change and the y-values are the same.
Stretch vs Shrink (Compression) - Horizontal and Vertical
Stretch vs Shrink:
Vertical Stretch vs Vertical Shrink (Compression)
On the top left we have a vertical shrink or compression. Our original function, f(x) - the red one, is taller than the new function, c *f(x) - the green one. For the new functions to be shorter, our c value needs to be 0 < c < 1. This make sense when we multiply our output value by a fraction that is less than 1 it will be shorter. Since the c value is less than 1, we need to flip the fraction when we write the number in the equation as we have by a factor of .... Another way of thinking about is. We want a graph that is smaller. How do we make things smaller? Subtraction or division. Let's divided by c. Since we are dividing by a fraction, we have 1/c. So, we need to flip the fraction and multiply by the reciprocal.
On the top right we have a vertical stretch. Our original function, f(x) - the red one, is the shorter one than the new function, c *f(x) - the green one. For the new functions to be taller, our c value needs to be c > 1. This make sense when we multiply our output value by a fraction that is greater than 1 it will be to be taller.
Notice how the c is outside the function. This affects the output value or the y-value, last.
Horizontal Stretch vs Horizontal Shrink (Compression)
On the bottom left we have a horizontal shrink or compression. Our original function, f(x) - the red one, is wider than the new function, f(c * x) - the green one. For the new functions to be narrower, our c value needs to be c > 1. This one sometimes takes a little think to get your head around it. When a horizontal shrink happens, the output values happen faster so our graph is compressed inward.
On the bottom right we have a horizontal stretch. Our original function, f(x) - the red one, is narrower than the new function, f(c * x) - the green one. For the new functions to be wider, our c value needs to be 0 < c < 1. This one sometimes takes a little think to get your head around it, as well. When a stretch happens, the output values happen slower so our graph is pulled outward. To do that we need divide by our factor or multiply by the reciprocal. We multiply by the reciprocal so our input values have a lower value, which means our output values have a lower value.
Notice how the c is inside the function. This affects the input value or the x-value, first.
The way I remember which one to flip is to think about the vertical shrink. That one has more common sense to it and remember that the horizontal one is the opposite.
Reflection Over the Axes
Reflection Over the Axes:
Reflection Over the X-axis
On the left we have a reflection over the x-axis. When we reflect over the x-axis we change the output since the x-values remain the same. So we have f(x) = -f(x).
Reflection Over the Y-axis
On the right we have a reflection over the y-axis. When we reflect over the y-axis we change the input since the y-values remain the same. So we have f(x) = f(-x).
Example 1
Example 1:
The first step is to graph our original function, f(x) = x2 - the black graph.
The second step that we are going to add is the horizontal shrink by a factor of 2 - the blue graph.
Our equation went from, f(x) = x2 => f(x) = (2x)2 .
Remember horizontal deals with the x-values so our factor of goes inside the parenthesis. Since we want our graph to shrink horizontally or compress inward, we need our output values to happen faster in this case the output value of 4.
Let's check our coordinate points to see if we have the right equation for our second step. We have (2, 4) for our original equation and (1, 4) for the new equation. The only change is the horizontal points, which is what we want. Now let's compare our x-values. Since we have a shrink we can divide our original value 2 by the factor 2 and the result is 1. That value matches the graph.
Now let's move on to the next step.
Now, we have the function, f(x) = (2x)2.
The next step that we are going to add is the vertical shrink by a factor of 8 - the orangish graph.
Our equation went from, f(x) = (2x)2 => f(x) = (1/8) * (2x)2 .
Remember vertical deals with the y-values so our factor of goes outside the parenthesis. Since we want our graph to shrink vertically or compress downward, we need our output values to happen slower in this case the output value of 1.
Let's check our coordinate points to see if we have the right equation for our third step. We have (1, 4) for blue equation and (1, 0.5) for the orangish equation. The only change is the vertical points, which is what we want. Now let's compare our y-values. Since we have a shrink we can divide our original value 4 by the factor 8 and the result is 0.5. That value matches the graph so we wrote down the new equation correctly.
Now let's move on to the next step.
Now, we have the function, f(x) = (2x)2.
The next step that we are going to add is the translation 1 down - the purple graph.
Our equation went from, f(x) = 1/8) * (2x)2 => f(x) = (1/8) * (2x)2 -1.
Remember translating deals with the y-values so our value goes outside the parenthesis. Since we want our graph to translate down, we need our output values to decrease by 1. So, we need to subtract by 1.
Let's check our coordinate points to see if we have the right equation for our fourth step. We have (0, 0) for orangish equation and (0, -1) for the purple equation. The only change is the vertical points, which is what we want. Now let's compare our y-values. Since we have translation 1 downward let's subtract 0 by 1 and the result is -1. That value matches the graph so we wrote down the new equation correctly.
Example 2
Example 2:
The first step is to graph our original function, f(x) = sqrt(x) - the black graph.
The second step that we are going to add is the horizontal stretch by a factor of 6 - the blue graph.
Our equation went from, f(x) = sqrt(x) => f(x) = sqrt([1/6] * x) .
Remember horizontal deals with the x-values so our factor of goes inside the parenthesis.
Since we want our graph to shrink vertically or compress downward, we need our output values to happen slower in this case the output value of 1.
Since we want our graph to stretch horizontally, we need to flip our factor so our output values happen slower in this case 1.
Let's check our coordinate points to see if we have the right equation for our second step. We have (1, 1) for our original equation and (6, 1) for the new equation. The only change is the horizontal points, which is what we want. Now let's compare our x-values. Since we have a stretch we can multiply our original value 1 (the x-value) by the factor 6 and the result is 6. That value matches the graph so we wrote down the new equation correctly.
Now let's move on to the next step.
The third step that we are going to add is the vertical stretch by a factor of 3 - the orangish graph.
Our equation went from, f(x) = sqrt([1/6] * x) => f(x) = 3 * sqrt([1/6] * x) .
Remember vertical deals with the y-values so our factor of goes outside the parenthesis. Since we want our graph to stretch vertically, we need our output values to happen faster in this case the output value of 3.
Let's check our coordinate points to see if we have the right equation for our third step. We have (6, 1) for blue equation and (6, 3) for the orangish equation. The only change is the vertical points, which is what we want. Now let's compare our y-values. Since we have a stretch we can multiply our original value 1 by the factor 3 and the result is 3. That value matches the graph so we wrote down the new equation correctly.
Now let's move on to the next step.
Now, we have the function, f(x) = 3 * sqrt([1/6] * x).
The next step that we are going to add is reflecting over the x-axis.
Our equation went from, f(x) = 3 * sqrt([1/6] * x) => f(x) = -3 * sqrt([1/6] * x).
Remember over the x-axis changes the final result or the output values so the negative goes outside the square root.
Let's check our coordinate points to see if we have the right equation for our fourth step. We have (6, 3) for orangish equation and (6, -3) for the purple equation. The only change is the vertical points, which is what we want. Now let's compare our y-values. Since we have a reflection over the x-axis, the original value is the negative of whatever value, and the negative of +3 is -3. That value matches the graph so we wrote down the new equation correctly.