Problem 54 Use transformations to graph eac... [FREE SOLUTION]

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Chapter 2: Problem 54

Use transformations to graph each function and state the domain and range. $$y=3|x|-200$$

Short Answer

Expert verified

Domain: $(-\infty, \infty)$; Range: $[-200, \infty)$.

Step by step solution

Identify the Parent Function

The parent function for this problem is the absolute value function: $y = |x|$.

Apply the Vertical Stretch

The coefficient 3 in front of $|x|$ indicates a vertical stretch by a factor of 3. This means the graph is stretched vertically, making it steeper.

Apply the Vertical Shift

The term $-200$ means the graph is shifted 200 units downward. This affects the y-values of the function.

Write the Transformed Function

Combining these transformations, the transformed function is $y = 3|x| - 200$.

Determine the Domain

The domain of an absolute value function is all real numbers, because you can input any x-value into $y = |x|$. Therefore, the domain is: $ \text{Domain} = (-\infty, \infty) $.

Determine the Range

The range is based on the y-values. Since the graph shifts down by 200 units, the lowest point on the graph is -200 and it extends upwards to infinity. Therefore, the range is: $ \text{Range} = [-200, \infty) $.

Graph the Function

Start with the basic graph of $y = |x|$, apply the vertical stretch to make it steeper, and then shift the entire graph down by 200 units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value function

The absolute value function is written as $ y = |x| $. This function forms a V-shaped graph that opens upwards.
The absolute value of a number is always positive or zero, so the graph never touches or goes below the x-axis.
For any input $ x $, the output is $ |x| $, which is the distance of $ x $ from zero on the number line.
This distance is always non-negative.
The general form of the absolute value function is simple but becomes more complex when combined with transformations.

vertical stretch

A vertical stretch changes the steepness of the graph without altering the shape of the function.
For the given exercise, the function $ y = 3|x| $ indicates a vertical stretch.
The '3' is a factor that multiplies each y-value by 3, making the V-shape steeper.
To visualize:

If you input $ x = 1 $, originally $ y = |1| = 1 $.
With the stretch: $ y = 3 |1| = 3 $.
If you input $ x = 2 $, originally $ y = |2| = 2 $.
With the stretch: $ y = 3 |2| = 6 $.

This means every point on the graph is stretched vertically, moving further away from the x-axis.

vertical shift

A vertical shift moves the whole graph up or down along the y-axis.
In the function $ y = 3|x| - 200 $, the '-200' indicates a vertical shift downward by 200 units.
This transformation affects all y-values, translating them 200 units down.
For instance:

Originally at $ x = 0 $, $ y = |0| = 0 $. With -200 shift: $ y = 0 - 200 = -200 $.
At $ x = 1 $, $ y = 3|1| = 3 $. Applying the shift: $ y = 3 - 200 = -197 $.

Thus, the whole function moves down, effectively lowering the vertex from $ (0, 0) $ to $ (0, -200) $.

domain

The domain of a function describes all possible input values (x-values) that can be used.
For the absolute value function $ y = |x| $, any real number can be an input.
This is true for all transformations of the absolute value function as well.
Thus, for $ y = 3|x| - 200 $, you can input any real number for x.
Therefore, the domain is:
Domain: $ (-fty, fty) $.

range

The range of a function identifies the set of all possible output values (y-values).
For an absolute value function like $ y = |x| $, the graph starts at the minimum y-value of 0 and goes upwards to infinity.
Including a transformation like $ y = 3|x| - 200 $, where the graph shifts downward by 200 units.
The new minimum y-value is now -200, transforming the range.
Therefore, the range is:
Range: $ [-200, fty) $.

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