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QuestionAnalyze the function f(x)=−2∣x∣f(x)=-2|x|. Find its anchor point, domain, range, intervals of increase/decrease, and intercepts.

Studdy Solution

STEP 1

Assumptions1. The function given is f(x)=−∣x∣f(x)=-|x| . The function is a transformation of the absolute value function, ∣x∣|x|, which is reflected over the x-axis and stretched vertically by a factor of.

STEP 2

The "Anchor Point" is the point where the graph of the function changes direction. For the absolute value function, this is at the origin (0,0).
Anchor Point=(0,0)Anchor\, Point = (0,0)

STEP 3

The domain of a function is the set of all possible x-values. Since the absolute value function is defined for all real numbers, the domain of f(x)=−2∣x∣f(x)=-2|x| is also all real numbers.
Domain=(−∞,∞)Domain = (-\infty, \infty)

STEP 4

The range of a function is the set of all possible y-values. Since the function f(x)=−2∣x∣f(x)=-2|x| is always less than or equal to0, the range is all real numbers less than or equal to0.
Range=(−∞,0]Range = (-\infty,0]

STEP 5

The function is increasing when the y-values increase as the x-values increase. Since f(x)=−2∣x∣f(x)=-2|x| is always decreasing or constant, there are no intervals where the function is increasing.
Increasing Intervals=∅Increasing\, Intervals = \emptyset

STEP 6

The function is decreasing when the y-values decrease as the x-values increase. Since f(x)=−2∣x∣f(x)=-2|x| is always decreasing or constant, the function is decreasing for all x-values not equal to0.
Decreasing Intervals=(−∞,0)∪(0,∞)Decreasing\, Intervals = (-\infty,0) \cup (0, \infty)

STEP 7

The x-intercept(s) is/are the x-value(s) where the function equals0. For the function f(x)=−2∣x∣f(x)=-2|x|, this occurs at x=0.
x−intercept=0x-intercept =0

STEP 8

The y-intercept is the y-value where the function crosses the y-axis. For the function f(x)=−2∣x∣f(x)=-2|x|, this occurs at y=0.
y−intercept=0y-intercept =0In conclusion, for the function f(x)=−2∣x∣f(x)=-2|x|- Anchor Point is at (0,0) - Domain is all real numbers- Range is all real numbers less than or equal to0- There are no intervals where the function is increasing- The function is decreasing for all x-values not equal to0- The x-intercept is at x=0- The y-intercept is at y=0

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