Chapter Review
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Start with a reflection across the x-axis.
Transformations: Reflection across the x-axis, vertical stretch by a factor of 3, translation 2 units to the left, and translation 1 unit down.
Graph:
We want to describe how to transform the parent function f(x)=x^2 to the graph of the given quadratic function. g(x)=- 3(x+2)^2-1 To do so, we need to consider four possible transformations.
Let's consider them one at the time.
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.
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 - 3. Therefore, the previous graph will be vertically stretched by a factor of 3.
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, 2 is being added to x, so the previous graph will be translated 2 units to the left.
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, 1 is subtracted from the whole function, so the previous graph will be translated 1 unit down.
Let's now graph the given function and the parent function f(x)=x^2 on the same coordinate grid.
Finally, let's summarize how to draw the graph of the given function when starting with the parent function, f(x)=x^2.