
Concept explainers
In Problems 21-32, graph the function by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). Verify your results using a graphing utility.
[ Hint: If necessary, write in the form .]

To graph: The transformation of the function .
Answer to Problem 27AYU
Explanation of Solution
Given:
We have to graph the function .
Graph:
The graph of the given function is
Interpretation:
The graph of is
Now, we can graph the given function by transforming the above graph.
The given function is .
The general form of a quadratic function is .
Here, if , the graph opens upward otherwise the graph opens downwards.
If is closer to 0, then the graph is shorter and wider.
If is large, then the graph is tall and narrow.
Then, the graph of is the graph of with units shifted horizontally and units shifted vertically.
Here, we can convert the given equation into the standard form by using the square completion technique.
The given equation can also be written as
Therefore, let us now add and subtract the square of half of the in the given equation.
Therefore, we get
Thus, the given equation can be written in the standard form as .
Since, is positive, the graph opens upwards.
We can see that is larger, therefore the graph is tall and narrow.
Here, the value of is twice the value of the function .
In the given function, we have and , therefore, the graph is the graph of with 1 units shifted horizontally right and 1 units shifted vertically downwards.
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