The graph will be a transformed version of the graph of y=x^2.
B
b
The graph will be a transformed version of the graph of y=x^3.
C
c
The graph will be a transformed version of the graph of y=|x|.
D
d
The graph will be a transformed version of the graph of y=sqrt(x).
A
a
B
b
C
c
D
d
Practice makes perfect
a
We want to graph the following equation using the parent graph.
y=-2(x- 3)^2+ 4
We can see that it is a quadratic equation. This means that its parent graph is y=x^2. To transform the parent graph to the given equation, we should consider four possible transformations.
Vertical translationsLet's consider them one at the time.
Reflection
Whenever x^2 is multiplied by a negative number, we will start by reflecting the graph across the x-axis.
Note how each x-coordinate stays the same, and how each y-coordinate changes its sign.
Stretch or Compression
We have a vertical stretch when x^2 is multiplied by a number whose absolute value is greater than one. If x^2 is multiplied by a number whose absolute value is less than one, a vertical compression will take place.
If x^2 is being multiplied by a negative number, the above still applies but everything will be upside down. In the given exercise, x^2 is multiplied by -2. Therefore, the previous graph will be vertically stretched by a factor of 2.
Horizontal Translation
If an addition or subtraction is applied to only the x-variable, the graph will be horizontally translated. In case of addition, the graph will be translated to the left. In case of subtraction, it will be moved to the right. In the given equation, 3 is being subtracted from x, so the previous graph will be translated three units to the right.
Vertical Translation
If an addition or subtraction is applied to the whole function, the graph will be vertically translated. In the case of addition, the graph will be translated up. In the case of subtraction, it will be moved downwards. In the given equation, 4 is added to the whole function, so the previous graph will be translated four units up.
Final Graph
Finally, we obtained the given equation.
b
Similarly, we want to graph the following equation using the transformations of the parent graph.
y=1/2(x+ 2)- 3
Since it is a cubic equation, its parent graph is y=x^3.
First, we can multiply the y-coordinates by 12. This shrinks the parent graph by a factor of 12.
Now, we will translate the graph 2 units left by subtracting 2 from each of the x-coordinates.
For the last transformation, we will translate the graph 3 units down. To do this, we subtract 3 from each y-coordinate.
Finally, we have the graph of the given equation.
c
Once again, let analyze the given equation.
y=2|x- 2|
This is an absolute value equation, so its parent graph is y=|x|.
First, we can multiply the y-coordinates by 2. This stretches the parent graph by a factor of 2.
Next, we will translate the graph 5 units right by adding 5 to each of the x-coordinates.
Finally, we obtained the graph of the given equation.
d
Let's analyze the last given equation.
y=sqrt(x- 2)- 3
Since it is a square root equation, its graph will be a transformed version of the parent graph, y=sqrt(x).
First, we can translate the graph 2 units to the right by adding 2 to each of the x-coordinates.
Now, we will translate the graph 3 units down by subtracting 3 from each of the y-coordinates.
