Question
Which phrases describe the graph of f(x) = |x| ? Check all that apply.
V-shaped
U-shaped
opens up
opens down
symmetric with respect to the x-axis
symmetric with respect to the y-axis
V-shaped
U-shaped
opens up
opens down
symmetric with respect to the x-axis
symmetric with respect to the y-axis
Answers
Answer
to prevent spam here is the answer. Complete the statements for the graph of f(x) = |x|.
The domain of the function is
✔ {x |x is all real numbers}
.
The range of the function is
✔ {y | y is greater than or equal to zero}
.
The graph is
✔ increasing
over the interval (0, ∞).
The graph is
✔ decreasing
over the interval (–∞, 0).
this other one is What are the coordinates of the vertex of the graph?
The vertex is
✔ (–2, 1)
.
now The vertex of the graph of f(x) = |x – 3| + 6 is located at (__,__).
The domain of the function is
✔ {x |x is all real numbers}
.
The range of the function is
✔ {y | y is greater than or equal to zero}
.
The graph is
✔ increasing
over the interval (0, ∞).
The graph is
✔ decreasing
over the interval (–∞, 0).
this other one is What are the coordinates of the vertex of the graph?
The vertex is
✔ (–2, 1)
.
now The vertex of the graph of f(x) = |x – 3| + 6 is located at (__,__).
Answer
On a coordinate plane, an absolute value graph has a vertex at (2.5, 1.5).The graph shows f(x) = |x – h| + k. What is the value of h?
h = 2.5
h = –1.5
h = 1.5
h = –2.5
h = 2.5
h = –1.5
h = 1.5
h = –2.5
Answer
On a coordinate plane, an absolute value graph has a vertex at (negative 1, negative 3).Which function is represented by the graph?
f(x) = |x – 1| + 3
f(x) = |x + 1| – 3
f(x) = |x – 1| – 3
f(x) = |x + 1| + 3
f(x) = |x – 1| + 3
f(x) = |x + 1| – 3
f(x) = |x – 1| – 3
f(x) = |x + 1| + 3
Answer
On which interval is the function increasing?
(–∞, –4)
(–∞, 4)
(–4, ∞)
(4, ∞)
x is -2 and 4
y is 1 and 2
(–∞, –4)
(–∞, 4)
(–4, ∞)
(4, ∞)
x is -2 and 4
y is 1 and 2
Answer
4 infinite yes
What are the domain and range of f(x) = |x – 3 | + 6?
Domain: {x | x is all real numbers}
Range: {y | y ≥ 6}
Domain: {x | x ≥ 3}
Range: {y | y ≥ 6}
Domain: {x | x is all real numbers}
Range: {y | y ≥ – 6}
Domain: {x | x ≥ 3}
Range: {y | y ≥ – 6}
What are the domain and range of f(x) = |x – 3 | + 6?
Domain: {x | x is all real numbers}
Range: {y | y ≥ 6}
Domain: {x | x ≥ 3}
Range: {y | y ≥ 6}
Domain: {x | x is all real numbers}
Range: {y | y ≥ – 6}
Domain: {x | x ≥ 3}
Range: {y | y ≥ – 6}
Answered by
GPT-4o mini
The graph of the function \( f(x) = |x| \) has certain characteristics. Here are the phrases that describe the graph correctly:
- **V-shaped**: True. The graph of \( f(x) = |x| \) forms a "V" shape because it has a vertex at the origin (0, 0) and extends upward in both directions.
- **U-shaped**: False. The graph is not U-shaped; it is distinctly "V" shaped.
- **Opens up**: True. The graph opens upwards, as both arms of the "V" extend upward as you move away from the vertex.
- **Opens down**: False. The graph does not open downward.
- **Symmetric with respect to the x-axis**: False. The graph is not symmetric with respect to the x-axis because it does not reflect over the x-axis.
- **Symmetric with respect to the y-axis**: True. The graph is symmetric with respect to the y-axis, meaning that if you fold the graph along the y-axis, the two sides will match.
So, the correct descriptions are:
- V-shaped
- Opens up
- Symmetric with respect to the y-axis
- **V-shaped**: True. The graph of \( f(x) = |x| \) forms a "V" shape because it has a vertex at the origin (0, 0) and extends upward in both directions.
- **U-shaped**: False. The graph is not U-shaped; it is distinctly "V" shaped.
- **Opens up**: True. The graph opens upwards, as both arms of the "V" extend upward as you move away from the vertex.
- **Opens down**: False. The graph does not open downward.
- **Symmetric with respect to the x-axis**: False. The graph is not symmetric with respect to the x-axis because it does not reflect over the x-axis.
- **Symmetric with respect to the y-axis**: True. The graph is symmetric with respect to the y-axis, meaning that if you fold the graph along the y-axis, the two sides will match.
So, the correct descriptions are:
- V-shaped
- Opens up
- Symmetric with respect to the y-axis
Answered by
GPT-4o mini
To find the vertex of the graph of the function \( f(x) = |x - 3| + 6 \), we can analyze the function's form.
The vertex form of the absolute value function \( f(x) = |x - h| + k \) indicates that the vertex is at the point \( (h, k) \).
In this case:
- \( h = 3 \) (the value that makes the expression inside the absolute value zero)
- \( k = 6 \) (the constant that is added, which shifts the graph vertically)
Therefore, the vertex of the graph \( f(x) = |x - 3| + 6 \) is located at:
**The vertex is** \( (3, 6) \).
The vertex form of the absolute value function \( f(x) = |x - h| + k \) indicates that the vertex is at the point \( (h, k) \).
In this case:
- \( h = 3 \) (the value that makes the expression inside the absolute value zero)
- \( k = 6 \) (the constant that is added, which shifts the graph vertically)
Therefore, the vertex of the graph \( f(x) = |x - 3| + 6 \) is located at:
**The vertex is** \( (3, 6) \).
Answered by
GPT-4o mini
In the vertex form of the absolute value function \( f(x) = |x - h| + k \), the vertex is given by the point \( (h, k) \).
Given that the vertex of the graph is at \( (2.5, 1.5) \), we can identify:
- \( h = 2.5 \)
- \( k = 1.5 \)
Thus, the value of \( h \) is:
**h = 2.5**.
Given that the vertex of the graph is at \( (2.5, 1.5) \), we can identify:
- \( h = 2.5 \)
- \( k = 1.5 \)
Thus, the value of \( h \) is:
**h = 2.5**.
Answered by
GPT-4o mini
To find the function that corresponds to the given vertex at \((-1, -3)\), we can use the vertex form of an absolute value function, which is:
\[
f(x) = |x - h| + k
\]
where \((h, k)\) is the vertex of the graph.
Here, with the vertex given as \((-1, -3)\):
- \(h = -1\)
- \(k = -3\)
Substituting these values into the vertex form gives us:
\[
f(x) = |x - (-1)| - 3 = |x + 1| - 3
\]
Thus, the correct function represented by the graph is:
**f(x) = |x + 1| - 3**.
\[
f(x) = |x - h| + k
\]
where \((h, k)\) is the vertex of the graph.
Here, with the vertex given as \((-1, -3)\):
- \(h = -1\)
- \(k = -3\)
Substituting these values into the vertex form gives us:
\[
f(x) = |x - (-1)| - 3 = |x + 1| - 3
\]
Thus, the correct function represented by the graph is:
**f(x) = |x + 1| - 3**.
Answered by
GPT-4o mini
To determine the intervals on which a piecewise function is increasing, we need to analyze the graph based on the given information about the points. Given points \((-2, 1)\) and \((4, 2)\) suggest that the function values are increasing from \(y = 1\) to \(y = 2\) as \(x\) goes from \(-2\) to \(4\).
However, we also have intervals to evaluate. Since we don't have the specific equation of the function, we'll use the given intervals and points. Typically, for an absolute value function or piecewise function defined by multiple sections, the graph will have specific intervals where it increases or decreases.
1. **From \(-\infty\) to \(-4\)**: This part may be decreasing.
2. **From \(-\infty\) to \(4\) includes up to \((-2, 1)\)**: A portion here could be increasing or decreasing, but since \(-2\) to \(4\) is increasing based on the given points.
3. **From \(-4\) to \(\infty\)**: Without knowing the function's behavior at \(-4\), we cannot definitively state whether this is increasing or decreasing.
4. **From \(4\) to \(\infty\)**: If the function continues to increase after \(4\), then this interval is also increasing.
Considering that you have points indicating that between \(-2\) and \(4\) the function is increasing, we can conclude:
- The function is increasing on the interval **\((-2, 4)\)**, but this interval is not provided in your options.
Given only the options provided, if the function behaves similarly in segments, **(4, ∞)** represents that it's increasing after \(4\) if we assume that’s the behavior. Thus:
The answer would likely be \( (4, ∞) \) based on the given options, provided that the function is indeed still increasing after \(4\). If the exact behavior of the function in relation to those intervals is unknown, I recommend checking the function's behavior at endpoints or additional intervals beyond those specified.
However, we also have intervals to evaluate. Since we don't have the specific equation of the function, we'll use the given intervals and points. Typically, for an absolute value function or piecewise function defined by multiple sections, the graph will have specific intervals where it increases or decreases.
1. **From \(-\infty\) to \(-4\)**: This part may be decreasing.
2. **From \(-\infty\) to \(4\) includes up to \((-2, 1)\)**: A portion here could be increasing or decreasing, but since \(-2\) to \(4\) is increasing based on the given points.
3. **From \(-4\) to \(\infty\)**: Without knowing the function's behavior at \(-4\), we cannot definitively state whether this is increasing or decreasing.
4. **From \(4\) to \(\infty\)**: If the function continues to increase after \(4\), then this interval is also increasing.
Considering that you have points indicating that between \(-2\) and \(4\) the function is increasing, we can conclude:
- The function is increasing on the interval **\((-2, 4)\)**, but this interval is not provided in your options.
Given only the options provided, if the function behaves similarly in segments, **(4, ∞)** represents that it's increasing after \(4\) if we assume that’s the behavior. Thus:
The answer would likely be \( (4, ∞) \) based on the given options, provided that the function is indeed still increasing after \(4\). If the exact behavior of the function in relation to those intervals is unknown, I recommend checking the function's behavior at endpoints or additional intervals beyond those specified.
Answered by
GPT-4o mini
To determine the domain and range of the function \( f(x) = |x - 3| + 6 \), let's analyze each part:
### Domain
The absolute value function \( |x - 3| \) is defined for all real numbers. Thus, the domain of \( f(x) \) is:
- **Domain:** {x | x is all real numbers}
### Range
The expression \( |x - 3| \) gives a minimum value of 0 (when \( x = 3 \)). Therefore:
- The minimum value of \( f(x) = |x - 3| + 6 \) occurs when \( |x - 3| = 0\), which is \( f(3) = 0 + 6 = 6 \).
- As \( x \) moves away from 3 in either direction, the value of \( f(x) \) increases without bound.
Thus, the range of \( f(x) \) is:
- **Range:** {y | y ≥ 6}
### Final Answer
Based on the above analysis, the correct pair is:
- **Domain:** {x | x is all real numbers}
- **Range:** {y | y ≥ 6}
### Domain
The absolute value function \( |x - 3| \) is defined for all real numbers. Thus, the domain of \( f(x) \) is:
- **Domain:** {x | x is all real numbers}
### Range
The expression \( |x - 3| \) gives a minimum value of 0 (when \( x = 3 \)). Therefore:
- The minimum value of \( f(x) = |x - 3| + 6 \) occurs when \( |x - 3| = 0\), which is \( f(3) = 0 + 6 = 6 \).
- As \( x \) moves away from 3 in either direction, the value of \( f(x) \) increases without bound.
Thus, the range of \( f(x) \) is:
- **Range:** {y | y ≥ 6}
### Final Answer
Based on the above analysis, the correct pair is:
- **Domain:** {x | x is all real numbers}
- **Range:** {y | y ≥ 6}
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