Question
On a coordinate plane, an curved line with a negative slope begins at (0, 0) and extends toward negative infinity.Which is the equation of the function in the graph?
Answers
Answer
so is it
( y = \sqrt{x} + |x| )
( y = \sqrt{x + |x|} )
( y = -\sqrt{x} + |x| )
( y = -(\sqrt{x} + |x|) )
( y = \sqrt{x} + |x| )
( y = \sqrt{x + |x|} )
( y = -\sqrt{x} + |x| )
( y = -(\sqrt{x} + |x|) )
Answer
just imagine the number in x and y i need to HOW youd put the equation to get the graph looking like 0,0 to infinity
Answer
okay the answer is y=[sqrtx+IxI]
Graph the function y = 4 square root of x. Then use the graph to find the missing x- or y-coordinates for the function to the nearest hundredth.
(4,
)
(5,
)
(
, 2.59)
Graph the function y = 4 square root of x. Then use the graph to find the missing x- or y-coordinates for the function to the nearest hundredth.
(4,
)
(5,
)
(
, 2.59)
Answered by
GPT-4o mini
To describe a curved line with a negative slope that starts at the point (0, 0) and extends toward negative infinity, we are likely looking for a function that is decreasing and possibly concave down in nature.
One type of function that fits this description is an exponential decay function, such as:
\[ f(x) = -e^{-x} \]
This function starts at (0, 0) because \( f(0) = -e^{0} = -1 \), which does not start at (0, 0), so we can modify it. Instead, a more suitable function could be:
\[ f(x) = -x^2 \]
This function has a downward-opening parabola, starts at (0, 0), and continues to negative infinity as \( x \) moves away from zero.
Another option could be a logarithmic function that also starts at zero and decreases towards negative infinity:
\[ f(x) = -\ln(x+1) \]
This function is undefined for \( x < -1 \), starts at the origin when \( x=0 \), and approaches negative infinity as \( x \) approaches \( -1 \) from the right.
In summary, functions like \( f(x) = -x^2 \) or \( f(x) = -\ln(x+1) \) may serve as candidates for a function that meets the criteria you described. Please specify any additional criteria if these do not meet the expectations.
One type of function that fits this description is an exponential decay function, such as:
\[ f(x) = -e^{-x} \]
This function starts at (0, 0) because \( f(0) = -e^{0} = -1 \), which does not start at (0, 0), so we can modify it. Instead, a more suitable function could be:
\[ f(x) = -x^2 \]
This function has a downward-opening parabola, starts at (0, 0), and continues to negative infinity as \( x \) moves away from zero.
Another option could be a logarithmic function that also starts at zero and decreases towards negative infinity:
\[ f(x) = -\ln(x+1) \]
This function is undefined for \( x < -1 \), starts at the origin when \( x=0 \), and approaches negative infinity as \( x \) approaches \( -1 \) from the right.
In summary, functions like \( f(x) = -x^2 \) or \( f(x) = -\ln(x+1) \) may serve as candidates for a function that meets the criteria you described. Please specify any additional criteria if these do not meet the expectations.
Answered by
GPT-4o mini
To analyze the options you provided, we need to determine whether each of these functions represents a curved line starting from (0, 0) and extending toward negative infinity with a negative slope.
1. **\( y = \sqrt{x} + |x| \)**:
- For \( x \geq 0 \): This simplifies to \( y = \sqrt{x} + x \), which is positive and increases as \( x \) increases.
- For \( x < 0 \): \( |x| = -x \), so the function is \( y = \sqrt{x} + (-x) \), which is not defined for \( x < 0 \).
- **Conclusion**: This function does not satisfy the conditions.
2. **\( y = \sqrt{x + |x|} \)**:
- For \( x \geq 0 \): This simplifies to \( y = \sqrt{x + x} = \sqrt{2x} \), which is also positive.
- For \( x < 0 \): This becomes \( y = \sqrt{x - x} = \sqrt{0} = 0 \), which gives \( y = 0 \).
- **Conclusion**: This function does not extend toward negative infinity and does not satisfy the conditions.
3. **\( y = -\sqrt{x} + |x| \)**:
- For \( x \geq 0 \): This simplifies to \( y = -\sqrt{x} + x \). As \( x \) increases, this expression could eventually grow positive, depending on the values.
- For \( x < 0 \): This becomes \( y = -\sqrt{x} + (-x) \), which is not defined \( ( \sqrt{x} \text{ is not defined for negative } x) \).
- **Conclusion**: This function does not meet the criteria.
4. **\( y = -(\sqrt{x} + |x|) \)**:
- For \( x \geq 0 \): This simplifies to \( y = -(\sqrt{x} + x) \), which is negative and decreases as \( x \) increases.
- For \( x < 0 \): This simplifies to \( y = -(\sqrt{x} - x) \), which is not defined since \( \sqrt{x} \) is not real for negative \( x \).
- **Conclusion**: This function is not defined for negative \( x \), so it doesn’t cover all criteria.
### Overall Conclusion:
None of the functions you provided fully meet the criteria of starting at (0, 0), having a negative slope, and extending toward negative infinity for all \( x \).
For functions that do fulfill those criteria, you might consider examples like:
- \( y = -x^2 \) (but only defined for \( x \leq 0 \)).
- \( y = -\ln(x + 1) \) (but only defined for \( x > -1 \)).
If you still want a particular form from the options, none seem to meet the criteria of a truly valid function that works for all \( x \). If you have any more specific requirements or contexts, please share!
1. **\( y = \sqrt{x} + |x| \)**:
- For \( x \geq 0 \): This simplifies to \( y = \sqrt{x} + x \), which is positive and increases as \( x \) increases.
- For \( x < 0 \): \( |x| = -x \), so the function is \( y = \sqrt{x} + (-x) \), which is not defined for \( x < 0 \).
- **Conclusion**: This function does not satisfy the conditions.
2. **\( y = \sqrt{x + |x|} \)**:
- For \( x \geq 0 \): This simplifies to \( y = \sqrt{x + x} = \sqrt{2x} \), which is also positive.
- For \( x < 0 \): This becomes \( y = \sqrt{x - x} = \sqrt{0} = 0 \), which gives \( y = 0 \).
- **Conclusion**: This function does not extend toward negative infinity and does not satisfy the conditions.
3. **\( y = -\sqrt{x} + |x| \)**:
- For \( x \geq 0 \): This simplifies to \( y = -\sqrt{x} + x \). As \( x \) increases, this expression could eventually grow positive, depending on the values.
- For \( x < 0 \): This becomes \( y = -\sqrt{x} + (-x) \), which is not defined \( ( \sqrt{x} \text{ is not defined for negative } x) \).
- **Conclusion**: This function does not meet the criteria.
4. **\( y = -(\sqrt{x} + |x|) \)**:
- For \( x \geq 0 \): This simplifies to \( y = -(\sqrt{x} + x) \), which is negative and decreases as \( x \) increases.
- For \( x < 0 \): This simplifies to \( y = -(\sqrt{x} - x) \), which is not defined since \( \sqrt{x} \) is not real for negative \( x \).
- **Conclusion**: This function is not defined for negative \( x \), so it doesn’t cover all criteria.
### Overall Conclusion:
None of the functions you provided fully meet the criteria of starting at (0, 0), having a negative slope, and extending toward negative infinity for all \( x \).
For functions that do fulfill those criteria, you might consider examples like:
- \( y = -x^2 \) (but only defined for \( x \leq 0 \)).
- \( y = -\ln(x + 1) \) (but only defined for \( x > -1 \)).
If you still want a particular form from the options, none seem to meet the criteria of a truly valid function that works for all \( x \). If you have any more specific requirements or contexts, please share!
Answered by
GPT-4o mini
To create a graph that starts at the point (0, 0) and extends toward negative infinity while having a negative slope, we can consider several mathematical functions that fulfill these criteria. Here's how to construct such a function:
### Function with Negative Slope
1. **Linear Function**:
- A simple linear function could be \( y = -x \).
- It starts at (0, 0) and continues downward toward negative infinity as \( x \) increases.
### Non-linear Functions
1. **Quadratic Function**:
- For a quadratic function that opens downward, such as \( y = -x^2 \):
- This also starts at (0, 0) and decreases towards negative infinity as \( x \) moves away from zero.
2. **Exponential Decay**:
- A function like \( y = -e^{-x} \):
- This function approaches (-1) as \( x \) increases but remains above -1, so it's an option for an asymptotic shape instead of strictly going toward negative infinity.
3. **Logarithmic Function**:
- A function such as \( y = -\ln(x + 1) \):
- This starts at (0, 0) and decreases as \( x \) increases. However, it only approaches negative infinity as \( x \) approaches -1 (not from the positive side).
### Choosing One Function
Given your requirement for a graph that "looks like (0, 0) to negative infinity," the quadratic function \( y = -x^2 \) is particularly desirable.
### How to Graph It:
1. **Linear Function**:
- Start by plotting the point (0, 0).
- Draw a straight line with a negative slope that crosses through the origin going downward.
2. **Quadratic Function**:
- Start at the point (0, 0).
- Draw a curve that opens downward, sitting at the vertex (the point at (0,0)) and starting to fall off sharply as you move away from the y-axis.
### Visualization
- The graph of \( y = -x^2 \) appears like an upside-down "U," while the graph of \( y = -x \) is simply a straight line on a downward slope.
### Summary
If you want the graph to start at (0, 0) and extend toward negative infinity, I recommend:
- **Linear Option**: \( y = -x \)
- **Non-linear Option**: \( y = -x^2 \)
These options will produce the desired effect on the coordinate plane.
### Function with Negative Slope
1. **Linear Function**:
- A simple linear function could be \( y = -x \).
- It starts at (0, 0) and continues downward toward negative infinity as \( x \) increases.
### Non-linear Functions
1. **Quadratic Function**:
- For a quadratic function that opens downward, such as \( y = -x^2 \):
- This also starts at (0, 0) and decreases towards negative infinity as \( x \) moves away from zero.
2. **Exponential Decay**:
- A function like \( y = -e^{-x} \):
- This function approaches (-1) as \( x \) increases but remains above -1, so it's an option for an asymptotic shape instead of strictly going toward negative infinity.
3. **Logarithmic Function**:
- A function such as \( y = -\ln(x + 1) \):
- This starts at (0, 0) and decreases as \( x \) increases. However, it only approaches negative infinity as \( x \) approaches -1 (not from the positive side).
### Choosing One Function
Given your requirement for a graph that "looks like (0, 0) to negative infinity," the quadratic function \( y = -x^2 \) is particularly desirable.
### How to Graph It:
1. **Linear Function**:
- Start by plotting the point (0, 0).
- Draw a straight line with a negative slope that crosses through the origin going downward.
2. **Quadratic Function**:
- Start at the point (0, 0).
- Draw a curve that opens downward, sitting at the vertex (the point at (0,0)) and starting to fall off sharply as you move away from the y-axis.
### Visualization
- The graph of \( y = -x^2 \) appears like an upside-down "U," while the graph of \( y = -x \) is simply a straight line on a downward slope.
### Summary
If you want the graph to start at (0, 0) and extend toward negative infinity, I recommend:
- **Linear Option**: \( y = -x \)
- **Non-linear Option**: \( y = -x^2 \)
These options will produce the desired effect on the coordinate plane.
Answered by
GPT-4o mini
To graph the function \( y = 4\sqrt{x} \) and find the missing x- or y-coordinates, we can follow these steps:
### Step 1: Understanding the Function
The function \( y = 4\sqrt{x} \) means that for every non-negative x-value, the output y-value will be four times the square root of that x-value.
### Step 2: Graphing the Function
1. **Plot Points**:
- Start with a few x-values to calculate corresponding y-values:
- If \( x = 0 \): \( y = 4\sqrt{0} = 0 \) ⇒ Point: (0, 0)
- If \( x = 1 \): \( y = 4\sqrt{1} = 4 \) ⇒ Point: (1, 4)
- If \( x = 4 \): \( y = 4\sqrt{4} = 8 \) ⇒ Point: (4, 8)
- If \( x = 9 \): \( y = 4\sqrt{9} = 12 \) ⇒ Point: (9, 12)
- If \( x = 16 \): \( y = 4\sqrt{16} = 16 \) ⇒ Point: (16, 16)
2. **Graph the Points**:
- Plot these points on the coordinate plane, and draw a smooth curve starting at (0, 0) and continuing upward and to the right, since the square root function increases.
### Step 3: Finding Missing Coordinates
Now, let's find the missing x- and y-coordinates for the given points on the graph.
1. **For the point (4, _)**:
\[
y = 4\sqrt{4} = 4 \times 2 = 8
\]
So, the missing coordinate is **(4, 8)**.
2. **For the point (5, _)**:
\[
y = 4\sqrt{5} \approx 4 \times 2.236 = 8.944
\]
So, the missing coordinate is **(5, 8.94)** (to the nearest hundredth).
3. **For the point (_, 2.59)**:
We need to determine x when \( y = 2.59 \):
\[
2.59 = 4\sqrt{x} \implies \sqrt{x} = \frac{2.59}{4} \implies \sqrt{x} = 0.6475
\]
Now square both sides:
\[
x = (0.6475)^2 \approx 0.4192
\]
So, the missing coordinate is **(0.42, 2.59)** (to the nearest hundredth).
### Summary of Missing Coordinates
- The coordinates you need are:
- \( (4, 8) \)
- \( (5, 8.94) \)
- \( (0.42, 2.59) \)
Feel free to ask if you need any further assistance!
### Step 1: Understanding the Function
The function \( y = 4\sqrt{x} \) means that for every non-negative x-value, the output y-value will be four times the square root of that x-value.
### Step 2: Graphing the Function
1. **Plot Points**:
- Start with a few x-values to calculate corresponding y-values:
- If \( x = 0 \): \( y = 4\sqrt{0} = 0 \) ⇒ Point: (0, 0)
- If \( x = 1 \): \( y = 4\sqrt{1} = 4 \) ⇒ Point: (1, 4)
- If \( x = 4 \): \( y = 4\sqrt{4} = 8 \) ⇒ Point: (4, 8)
- If \( x = 9 \): \( y = 4\sqrt{9} = 12 \) ⇒ Point: (9, 12)
- If \( x = 16 \): \( y = 4\sqrt{16} = 16 \) ⇒ Point: (16, 16)
2. **Graph the Points**:
- Plot these points on the coordinate plane, and draw a smooth curve starting at (0, 0) and continuing upward and to the right, since the square root function increases.
### Step 3: Finding Missing Coordinates
Now, let's find the missing x- and y-coordinates for the given points on the graph.
1. **For the point (4, _)**:
\[
y = 4\sqrt{4} = 4 \times 2 = 8
\]
So, the missing coordinate is **(4, 8)**.
2. **For the point (5, _)**:
\[
y = 4\sqrt{5} \approx 4 \times 2.236 = 8.944
\]
So, the missing coordinate is **(5, 8.94)** (to the nearest hundredth).
3. **For the point (_, 2.59)**:
We need to determine x when \( y = 2.59 \):
\[
2.59 = 4\sqrt{x} \implies \sqrt{x} = \frac{2.59}{4} \implies \sqrt{x} = 0.6475
\]
Now square both sides:
\[
x = (0.6475)^2 \approx 0.4192
\]
So, the missing coordinate is **(0.42, 2.59)** (to the nearest hundredth).
### Summary of Missing Coordinates
- The coordinates you need are:
- \( (4, 8) \)
- \( (5, 8.94) \)
- \( (0.42, 2.59) \)
Feel free to ask if you need any further assistance!
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