The graph of f(x) = |x| is reflected across the x-axis and translated right 8 units. Which statement about the transformed function is correct?

Question

The domain is the same as the parent function’s domain.
The graph opens in the same direction as the parent function.
The range is the same as the parent function’s range.
The vertex is the same as the parent function’s vertex.

Answer 1

To analyze the transformation of the function \( f(x) = |x| \), let's break down the steps involved:

1. Reflection across the x-axis: Reflecting \( f(x) = |x| \) across the x-axis results in the function \( g(x) = -|x| \). The original function opens upwards (has a V-shape), while the reflected function opens downwards.

2. Translation right by 8 units: Translating \( g(x) = -|x| \) to the right by 8 units results in the function \( h(x) = -|x - 8| \). This means the entire graph moves 8 units to the right.

Now let's evaluate each statement:

1. The domain is the same as the parent function’s domain.

   • The domain of \( f(x) = |x| \) is all real numbers \( (-\infty, \infty) \). The domain of \( h(x) = -|x - 8| \) is also all real numbers \( (-\infty, \infty) \). This statement is correct.

2. The graph opens in the same direction as the parent function.

   • The graph of \( h(x) = -|x - 8| \) opens downward (since it has a negative in front of the absolute value), while the parent function \( f(x) = |x| \) opens upward. This statement is incorrect.

3. The range is the same as the parent function’s range.

   • The range of the parent function \( f(x) = |x| \) is \( [0, \infty) \), while the range of the transformed function \( h(x) = -|x - 8| \) is \( (-\infty, 0] \) because the graph opens downward. This statement is incorrect.

4. The vertex is the same as the parent function’s vertex.

   • The vertex of \( f(x) = |x| \) is at (0, 0), whereas the vertex of \( h(x) = -|x - 8| \) is at (8, 0), since it has been translated right by 8 units. This statement is incorrect.

In summary, the correct statement about the transformed function is: The domain is the same as the parent function’s domain.

Answer 2

answer for this other question

=–|x – 2|

Which describes how the parent function, f(x) = |x|, is transformed to show the function
f(x) = 0.1|x – 3|?

It is wider and shifted 3 units to the left.
It is wider and shifted 3 units to the right.
It is narrower and shifted 3 units to the left.
It is narrower and shifted 3 units to the right.

Answer 3

To determine how the parent function \( f(x) = |x| \) is transformed to show the function \( f(x) = 0.1|x - 3| \), we need to analyze the transformations involved:

1. Vertical Compression: The coefficient \( 0.1 \) in front of the absolute value function indicates a vertical compression. This means the graph will be wider than the parent function \( f(x) = |x| \).

2. Horizontal Translation: The term \( x - 3 \) indicates a horizontal shift. Since it is \( (x - 3) \), this indicates a shift to the right by 3 units.

Putting the transformations together, we see that the function \( f(x) = 0.1|x - 3| \) results in a graph that is wider due to the vertical compression and shifted 3 units to the right.

Based on this analysis, the correct description of the transformation is:

It is wider and shifted 3 units to the right.

Answer 4

Which absolute value functions will be narrower than the parent function, f(x) = |x|? Check all that apply.

f(x) = |x|
f(x) = |x – 2|
f(x) = |x| + 3
f(x) = 2.9|x|
f(x) = 1.2|x + 8|
f(x) = 0.7|x| – 3.2

Answer 5

An absolute value function has the general form \( f(x) = a|x - h| + k \), where:

• \( |a| > 1 \) indicates a narrower graph than the parent function \( f(x) = |x| \).
• \( |a| < 1 \) indicates a wider graph than the parent function.

Now let's evaluate each function to determine whether it is narrower than the parent function \( f(x) = |x| \):

1. \( f(x) = |x| \):

   • This is the parent function itself.
   • Not narrower.

2. \( f(x) = |x – 2| \):

   • The slope (coefficient of the absolute value) is 1 (i.e., \( a = 1 \)).
   • Not narrower.

3. \( f(x) = |x| + 3 \):

   • The slope (coefficient of the absolute value) is 1 (i.e., \( a = 1 \)).
   • Not narrower.

4. \( f(x) = 2.9|x| \):

   • The slope (coefficient of the absolute value) is 2.9 (i.e., \( a = 2.9 \)).
   • Since \( 2.9 > 1 \), this function is narrower than the parent function.

5. \( f(x) = 1.2|x + 8| \):

   • The slope (coefficient of the absolute value) is 1.2 (i.e., \( a = 1.2 \)).
   • Since \( 1.2 > 1 \), this function is narrower than the parent function.

6. \( f(x) = 0.7|x| – 3.2 \):

   • The slope (coefficient of the absolute value) is 0.7 (i.e., \( a = 0.7 \)).
   • Since \( 0.7 < 1 \), this function is wider, not narrower.

Summary:

The absolute value functions that are narrower than the parent function \( f(x) = |x| \) are:

• \( f(x) = 2.9|x| \)
• \( f(x) = 1.2|x + 8| \)

These are the two functions that will be narrower than the parent function.

Answer 6

answer to a other question I DIDNT ASK BOT its just a question i have the answer to.
The graph of the function f(x) = 0.5|x| is
✔ wider
than its parent graph, f(x) = |x|.



The domain of the function is
✔ all real numbers
.



The range of the function is
✔ all real numbers greater than or equal to 0
.

and another answer i did not ask bot. f(x)= 1/3 |x-1|

also another one i didnt ask bot.
Determine the domain and range of the function f(x) = –|x| + 2.



The domain of the function is
✔ all real numbers
.

The range of the function is
✔ all numbers less than or equal to 2
.

another one –|x| + 3

NOW BOT

The graph of f(x) = |x| has been stretched by a factor of 2.5. If no other transformations of the function have occurred, which point lies on the new graph?

( –4, –4)
(–3, 7.5)
( –2, 5.5)
( –1, –2.5)

Answer 7

To find the point that lies on the new graph of the function \( f(x) \) after it has been stretched by a factor of 2.5, we start with the parent function \( f(x) = |x| \).

When the graph of \( f(x) = |x| \) is stretched vertically by a factor of 2.5, the new function becomes: \[ g(x) = 2.5|x| \]

Now, let's apply this transformation to each of the given points to see if they lie on the new graph \( g(x) \).

1. For the point \( (-4, -4) \): \[ g(-4) = 2.5|-4| = 2.5 \times 4 = 10 \] The point \( (-4, -4) \) does not lie on the graph since the \( y \)-value should be 10.

2. For the point \( (-3, 7.5) \): \[ g(-3) = 2.5|-3| = 2.5 \times 3 = 7.5 \] This point lies on the graph since the \( y \)-value matches.

3. For the point \( (-2, 5.5) \): \[ g(-2) = 2.5|-2| = 2.5 \times 2 = 5 \] The point \( (-2, 5.5) \) does not lie on the graph since the \( y \)-value should be 5.

4. For the point \( (-1, -2.5) \): \[ g(-1) = 2.5|-1| = 2.5 \times 1 = 2.5 \] The point \( (-1, -2.5) \) does not lie on the graph since the \( y \)-value should be 2.5.

Conclusion:

The only point that lies on the new graph after the vertical stretch by a factor of 2.5 is:

(–3, 7.5).

Answer 8

Which statements are true about the function f(x) = –|x| – 2? Check all that apply.

The parent function and this function open in the same direction.
The parent function and this function have the same range.
The parent function and this function have the same domain.
The range of the function is all real numbers less than or equal to –2.
The domain of the function is all real numbers greater than or equal to –2.
There are no values from the domain of the parent function that are also in the domain of this function.

Answer 9

To assess the function \( f(x) = -|x| - 2 \) and determine which statements are true, we can analyze the transformations made to the parent function \( f(x) = |x| \).

1. The parent function and this function open in the same direction.

   • The parent function \( f(x) = |x| \) opens upwards, while \( f(x) = -|x| - 2 \) opens downwards because of the negative sign in front of \( |x| \).
   • This statement is false.

2. The parent function and this function have the same range.

   • The range of the parent function \( f(x) = |x| \) is \( [0, \infty) \). The range of \( f(x) = -|x| - 2 \) is \( (-\infty, -2] \) (the output values go downwards and approach negative infinity).
   • This statement is false.

3. The parent function and this function have the same domain.

   • The domain of the parent function \( f(x) = |x| \) is all real numbers \( (-\infty, \infty) \). The domain of \( f(x) = -|x| - 2 \) is also all real numbers \( (-\infty, \infty) \).
   • This statement is true.

4. The range of the function is all real numbers less than or equal to –2.

   • As we established, the range of \( f(x) = -|x| - 2 \) is \( (-\infty, -2] \). Thus, it includes all real numbers less than or equal to -2.
   • This statement is true.

5. The domain of the function is all real numbers greater than or equal to –2.

   • This is incorrect as the domain is all real numbers (not limited to those greater than or equal to -2).
   • This statement is false.

6. There are no values from the domain of the parent function that are also in the domain of this function.

   • Since both the parent function and the transformed function have the same domain (all real numbers), there are indeed values from the domain of the parent function that are also in the domain of this function.
   • This statement is false.

Summary:

The true statements about the function \( f(x) = -|x| - 2 \) are:

• The parent function and this function have the same domain.
• The range of the function is all real numbers less than or equal to –2.