Question
Use the point-slope equation to identify the slope and the coordinates of a point
on the line y – 4 = (x – 1).
The slope of the line is
.
A point on the line is
.
on the line y – 4 = (x – 1).
The slope of the line is
.
A point on the line is
.
Answers
Answered by
GPT-4o mini
The point-slope equation is given in the form:
\[ y - y_1 = m(x - x_1) \]
where \(m\) is the slope and \((x_1, y_1)\) is a point on the line.
From the equation you provided:
\[ y - 4 = (x - 1) \]
We can rewrite this in the standard point-slope form. Here, the slope \(m\) is the coefficient of \(x\), which is \(1\) (since it can be inferred that the equation can also be represented as \(y - 4 = 1(x - 1)\)).
Therefore, the slope of the line is:
**The slope of the line is** \(1\).
To find a point on the line, we can use the point \((x_1, y_1)\) which is given in the equation as \( (1, 4) \).
**A point on the line is** \( (1, 4) \).
\[ y - y_1 = m(x - x_1) \]
where \(m\) is the slope and \((x_1, y_1)\) is a point on the line.
From the equation you provided:
\[ y - 4 = (x - 1) \]
We can rewrite this in the standard point-slope form. Here, the slope \(m\) is the coefficient of \(x\), which is \(1\) (since it can be inferred that the equation can also be represented as \(y - 4 = 1(x - 1)\)).
Therefore, the slope of the line is:
**The slope of the line is** \(1\).
To find a point on the line, we can use the point \((x_1, y_1)\) which is given in the equation as \( (1, 4) \).
**A point on the line is** \( (1, 4) \).
Answer
The slope of the line is
✔ m = 1/2
.
A point on the line is
✔ (1, 4)
.
Which point-slope equation represents a line that passes through (3, –2) with a slope of negative StartFraction 4 Over 5 EndFraction?
y – 3 =4/5 –(x + 2)
y – 2 =4/5 (x – 3)
y + 2 =4/5 –(x – 3)
y + 3 =4/5 (x + 2)
✔ m = 1/2
.
A point on the line is
✔ (1, 4)
.
Which point-slope equation represents a line that passes through (3, –2) with a slope of negative StartFraction 4 Over 5 EndFraction?
y – 3 =4/5 –(x + 2)
y – 2 =4/5 (x – 3)
y + 2 =4/5 –(x – 3)
y + 3 =4/5 (x + 2)
Answered by
GPT-4o mini
To find the point-slope equation that represents a line passing through the point \((3, -2)\) with a slope of \(-\frac{4}{5}\), we use the point-slope form:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point and \(m\) is the slope. Here, \(x_1 = 3\), \(y_1 = -2\), and \(m = -\frac{4}{5}\).
Plugging in the values:
\[ y - (-2) = -\frac{4}{5}(x - 3) \]
This simplifies to:
\[ y + 2 = -\frac{4}{5}(x - 3) \]
Now let's look at the options:
1. **\(y - 3 = \frac{4}{5} - (x + 2)\)** (Incorrect form)
2. **\(y - 2 = \frac{4}{5}(x - 3)\)** (Incorrect slope)
3. **\(y + 2 = \frac{4}{5} - (x - 3)\)** (Incorrect slope and form)
4. **\(y + 3 = \frac{4}{5}(x + 2)\)** (Incorrect slope and point)
None of these options directly correspond to our derived equation.
However, let's rewrite our final equation to check if it's one of the provided options. We have:
\[ y + 2 = -\frac{4}{5}(x - 3) \]
This means the slope is \(-\frac{4}{5}\), and we can rearrange it as:
\[ y + 2 = -\frac{4}{5}x + \frac{12}{5} \]
which rearranges to
\[ y = -\frac{4}{5}x + \frac{12}{5} - 2 \]
or
\[ y = -\frac{4}{5}x + \frac{2}{5} \]
None of the options correctly correspond to this equation.
There may be an error in the options provided if we are looking specifically for the point-slope equation. Based on our calculations, the correct point-slope form should be:
**\(y + 2 = -\frac{4}{5}(x - 3)\)**
This doesn't match any of the provided answers, indicating that none of the options represent the desired line through (3, -2) with a slope of -4/5. If you have more context or need to confirm the options, please let me know!
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point and \(m\) is the slope. Here, \(x_1 = 3\), \(y_1 = -2\), and \(m = -\frac{4}{5}\).
Plugging in the values:
\[ y - (-2) = -\frac{4}{5}(x - 3) \]
This simplifies to:
\[ y + 2 = -\frac{4}{5}(x - 3) \]
Now let's look at the options:
1. **\(y - 3 = \frac{4}{5} - (x + 2)\)** (Incorrect form)
2. **\(y - 2 = \frac{4}{5}(x - 3)\)** (Incorrect slope)
3. **\(y + 2 = \frac{4}{5} - (x - 3)\)** (Incorrect slope and form)
4. **\(y + 3 = \frac{4}{5}(x + 2)\)** (Incorrect slope and point)
None of these options directly correspond to our derived equation.
However, let's rewrite our final equation to check if it's one of the provided options. We have:
\[ y + 2 = -\frac{4}{5}(x - 3) \]
This means the slope is \(-\frac{4}{5}\), and we can rearrange it as:
\[ y + 2 = -\frac{4}{5}x + \frac{12}{5} \]
which rearranges to
\[ y = -\frac{4}{5}x + \frac{12}{5} - 2 \]
or
\[ y = -\frac{4}{5}x + \frac{2}{5} \]
None of the options correctly correspond to this equation.
There may be an error in the options provided if we are looking specifically for the point-slope equation. Based on our calculations, the correct point-slope form should be:
**\(y + 2 = -\frac{4}{5}(x - 3)\)**
This doesn't match any of the provided answers, indicating that none of the options represent the desired line through (3, -2) with a slope of -4/5. If you have more context or need to confirm the options, please let me know!
Answer
A coordinate grid with a line passing through the points (negative 4, negative 4), (0, negative 2), and (4, 0).What is the equation of the line in point-slope form?y + 4 =1/2 (x + 4)y – 4 =1/2 (x + 4)y – 0 = 2(x - 4)y – 4 = 2(x – 0)
Answered by
GPT-4o mini
To find the equation of the line passing through the points \((-4, -4)\), \((0, -2)\), and \((4, 0)\) in point-slope form, we first need to determine the slope of the line.
### Step 1: Calculate the slope
Using the points \((0, -2)\) and \((4, 0)\):
\[
\text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-2)}{4 - 0} = \frac{2}{4} = \frac{1}{2}
\]
### Step 2: Point-slope form
Now we can use point-slope form, which is given by:
\[
y - y_1 = m(x - x_1)
\]
We can use any of the three points for the equation; let’s use the point \( (0, -2) \):
\[
y - (-2) = \frac{1}{2}(x - 0)
\]
This simplifies to:
\[
y + 2 = \frac{1}{2}x
\]
### Selecting the correct answer
Now, let’s look at the provided options:
1. \(y + 4 = \frac{1}{2}(x + 4)\) (This does not match)
2. **\(y - 4 = \frac{1}{2}(x + 4)\)** (This does not match)
3. \(y - 0 = 2(x - 4)\) (This does not match)
4. \(y - 4 = 2(x - 0)\) (This does not match)
The slope we calculated is \(\frac{1}{2}\), not \(2\), and using the point \((0, -2)\) gives us \(y + 2 = \frac{1}{2}x\).
However, if we use the point \((-4, -4)\) in point-slope form with a slope of \(\frac{1}{2}\):
\[
y - (-4) = \frac{1}{2}(x + 4)
\]
This simplifies to:
\[
y + 4 = \frac{1}{2}(x + 4)
\]
This matches **option 1: \(y + 4 = \frac{1}{2}(x + 4)\)**.
Thus, the correct answer for the equation of the line in point-slope form is:
**\(y + 4 = \frac{1}{2}(x + 4)\)**.
### Step 1: Calculate the slope
Using the points \((0, -2)\) and \((4, 0)\):
\[
\text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-2)}{4 - 0} = \frac{2}{4} = \frac{1}{2}
\]
### Step 2: Point-slope form
Now we can use point-slope form, which is given by:
\[
y - y_1 = m(x - x_1)
\]
We can use any of the three points for the equation; let’s use the point \( (0, -2) \):
\[
y - (-2) = \frac{1}{2}(x - 0)
\]
This simplifies to:
\[
y + 2 = \frac{1}{2}x
\]
### Selecting the correct answer
Now, let’s look at the provided options:
1. \(y + 4 = \frac{1}{2}(x + 4)\) (This does not match)
2. **\(y - 4 = \frac{1}{2}(x + 4)\)** (This does not match)
3. \(y - 0 = 2(x - 4)\) (This does not match)
4. \(y - 4 = 2(x - 0)\) (This does not match)
The slope we calculated is \(\frac{1}{2}\), not \(2\), and using the point \((0, -2)\) gives us \(y + 2 = \frac{1}{2}x\).
However, if we use the point \((-4, -4)\) in point-slope form with a slope of \(\frac{1}{2}\):
\[
y - (-4) = \frac{1}{2}(x + 4)
\]
This simplifies to:
\[
y + 4 = \frac{1}{2}(x + 4)
\]
This matches **option 1: \(y + 4 = \frac{1}{2}(x + 4)\)**.
Thus, the correct answer for the equation of the line in point-slope form is:
**\(y + 4 = \frac{1}{2}(x + 4)\)**.
Answer
Which graph represents the function y – 3 = y minus 3 equals StartFraction 3 Over 2 EndFraction left-parenthesis x minus 4 right-parenthesis.(x – 4)?A coordinate grid with a line passing through the points (0, negative 3) and (2, 0).A coordinate grid with a line passing through the points (negative 2, 0) and (0, negative 3).A coordinate grid with a line passing through the points (negative 4, negative 3) and (2, 1).A coordinate grid with a line passing through the points (negative 1, 1) and (1, 1).
Answered by
GPT-4o mini
To determine which graph represents the function given by the equation:
\[
y - 3 = \frac{3}{2}(x - 4)
\]
### Step 1: Convert to slope-intercept form
Let's rewrite the equation into the slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
Starting from:
\[
y - 3 = \frac{3}{2}(x - 4)
\]
Distributing \(\frac{3}{2}\):
\[
y - 3 = \frac{3}{2}x - 6
\]
Now, add \(3\) to both sides to isolate \(y\):
\[
y = \frac{3}{2}x - 6 + 3
\]
This simplifies to:
\[
y = \frac{3}{2}x - 3
\]
### Step 2: Identify the slope and y-intercept
From this equation, we can see that:
- The slope \(m = \frac{3}{2}\)
- The y-intercept \(b = -3\)
This means the line crosses the y-axis at \((0, -3)\).
### Step 3: Determine another point on the line
Using the slope of \( \frac{3}{2} \), we can find another point on the line. Starting from the y-intercept \((0, -3)\):
From \((0, -3)\), moving to the right (positive \(x\) direction):
- If we increase \(x\) by \(2\) (the denominator of the slope), we can rise \(3\) (the numerator of the slope):
- Moving from \((0, -3)\) to \((2, 0)\) gives us the point \((2, 0)\).
### Step 4: Matching with the available options
Now, we need to see which option mentions the points \((0, -3)\) and \((2, 0)\):
1. **A coordinate grid with a line passing through the points (0, -3) and (2, 0)** (This is a match!)
2. A coordinate grid with a line passing through the points (-2, 0) and (0, -3).
3. A coordinate grid with a line passing through the points (-4, -3) and (2, 1).
4. A coordinate grid with a line passing through the points (-1, 1) and (1, 1).
### Conclusion
The graph that represents the function \(y - 3 = \frac{3}{2}(x - 4)\) is:
**A coordinate grid with a line passing through the points (0, -3) and (2, 0).**
\[
y - 3 = \frac{3}{2}(x - 4)
\]
### Step 1: Convert to slope-intercept form
Let's rewrite the equation into the slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
Starting from:
\[
y - 3 = \frac{3}{2}(x - 4)
\]
Distributing \(\frac{3}{2}\):
\[
y - 3 = \frac{3}{2}x - 6
\]
Now, add \(3\) to both sides to isolate \(y\):
\[
y = \frac{3}{2}x - 6 + 3
\]
This simplifies to:
\[
y = \frac{3}{2}x - 3
\]
### Step 2: Identify the slope and y-intercept
From this equation, we can see that:
- The slope \(m = \frac{3}{2}\)
- The y-intercept \(b = -3\)
This means the line crosses the y-axis at \((0, -3)\).
### Step 3: Determine another point on the line
Using the slope of \( \frac{3}{2} \), we can find another point on the line. Starting from the y-intercept \((0, -3)\):
From \((0, -3)\), moving to the right (positive \(x\) direction):
- If we increase \(x\) by \(2\) (the denominator of the slope), we can rise \(3\) (the numerator of the slope):
- Moving from \((0, -3)\) to \((2, 0)\) gives us the point \((2, 0)\).
### Step 4: Matching with the available options
Now, we need to see which option mentions the points \((0, -3)\) and \((2, 0)\):
1. **A coordinate grid with a line passing through the points (0, -3) and (2, 0)** (This is a match!)
2. A coordinate grid with a line passing through the points (-2, 0) and (0, -3).
3. A coordinate grid with a line passing through the points (-4, -3) and (2, 1).
4. A coordinate grid with a line passing through the points (-1, 1) and (1, 1).
### Conclusion
The graph that represents the function \(y - 3 = \frac{3}{2}(x - 4)\) is:
**A coordinate grid with a line passing through the points (0, -3) and (2, 0).**
Answer
if you're confused it's the line that goes right through -4 and starts at the right
Choose a linear function for the line represented by the point-slope equation y – 5 = 3(x – 2).
f(x) = 3x + 1
f(x) = 3x – 1
f(x) = 8x + 10
f(x) = 8x – 10
Choose a linear function for the line represented by the point-slope equation y – 5 = 3(x – 2).
f(x) = 3x + 1
f(x) = 3x – 1
f(x) = 8x + 10
f(x) = 8x – 10
Answered by
GPT-4o mini
To find the linear function represented by the point-slope equation \(y - 5 = 3(x - 2)\), we will convert it into slope-intercept form (\(y = mx + b\)).
### Step 1: Expand the point-slope equation
Starting with the equation:
\[
y - 5 = 3(x - 2)
\]
Distributing \(3\) on the right side:
\[
y - 5 = 3x - 6
\]
### Step 2: Solve for \(y\)
Now, we’ll add \(5\) to both sides to isolate \(y\):
\[
y = 3x - 6 + 5
\]
\[
y = 3x - 1
\]
### Step 3: Define the function
The resulting equation can be written as a function:
\[
f(x) = 3x - 1
\]
### Conclusion
Now, we compare this with the given options:
1. \(f(x) = 3x + 1\)
2. **\(f(x) = 3x - 1\)** (This is our match!)
3. \(f(x) = 8x + 10\)
4. \(f(x) = 8x - 10\)
The correct linear function for the line represented by the point-slope equation \(y - 5 = 3(x - 2)\) is:
**\(f(x) = 3x - 1\)**.
### Step 1: Expand the point-slope equation
Starting with the equation:
\[
y - 5 = 3(x - 2)
\]
Distributing \(3\) on the right side:
\[
y - 5 = 3x - 6
\]
### Step 2: Solve for \(y\)
Now, we’ll add \(5\) to both sides to isolate \(y\):
\[
y = 3x - 6 + 5
\]
\[
y = 3x - 1
\]
### Step 3: Define the function
The resulting equation can be written as a function:
\[
f(x) = 3x - 1
\]
### Conclusion
Now, we compare this with the given options:
1. \(f(x) = 3x + 1\)
2. **\(f(x) = 3x - 1\)** (This is our match!)
3. \(f(x) = 8x + 10\)
4. \(f(x) = 8x - 10\)
The correct linear function for the line represented by the point-slope equation \(y - 5 = 3(x - 2)\) is:
**\(f(x) = 3x - 1\)**.
Answer
A coordinate grid with a line passing through the points (negative 2, negative 1) and (2, 0)Use the graph of the line to answer the questions.What is an equation of the line in point-slope form?How can the point-slope form be written in function notation?
Answered by
GPT-4o mini
To find the equation of the line that passes through the points \((-2, -1)\) and \((2, 0)\), we first need to calculate the slope of the line using the slope formula:
### Step 1: Calculate the slope
The slope \(m\) can be calculated as follows:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-1)}{2 - (-2)} = \frac{0 + 1}{2 + 2} = \frac{1}{4}
\]
### Step 2: Use point-slope form
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
We can use either of the two points for this form. Let's use the point \((-2, -1)\):
\[
y - (-1) = \frac{1}{4}(x - (-2))
\]
This simplifies to:
\[
y + 1 = \frac{1}{4}(x + 2)
\]
### Equation in Point-Slope Form
Thus, the equation of the line in point-slope form is:
\[
y + 1 = \frac{1}{4}(x + 2)
\]
### Step 3: Write in Function Notation
To write this in function notation, we can express \(y\) as \(f(x)\):
Replace \(y\) with \(f(x)\):
\[
f(x) + 1 = \frac{1}{4}(x + 2)
\]
This can be rearranged to give:
\[
f(x) = \frac{1}{4}(x + 2) - 1
\]
Now simplify the function:
\[
f(x) = \frac{1}{4}x + \frac{2}{4} - 1
\]
\[
f(x) = \frac{1}{4}x + \frac{1}{2} - 1
\]
\[
f(x) = \frac{1}{4}x - \frac{1}{2}
\]
### Summary
1. **Equation in Point-Slope Form**: \(y + 1 = \frac{1}{4}(x + 2)\)
2. **Function Notation**: \(f(x) = \frac{1}{4}x - \frac{1}{2}\)
Feel free to ask if you need any further assistance!
### Step 1: Calculate the slope
The slope \(m\) can be calculated as follows:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-1)}{2 - (-2)} = \frac{0 + 1}{2 + 2} = \frac{1}{4}
\]
### Step 2: Use point-slope form
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
We can use either of the two points for this form. Let's use the point \((-2, -1)\):
\[
y - (-1) = \frac{1}{4}(x - (-2))
\]
This simplifies to:
\[
y + 1 = \frac{1}{4}(x + 2)
\]
### Equation in Point-Slope Form
Thus, the equation of the line in point-slope form is:
\[
y + 1 = \frac{1}{4}(x + 2)
\]
### Step 3: Write in Function Notation
To write this in function notation, we can express \(y\) as \(f(x)\):
Replace \(y\) with \(f(x)\):
\[
f(x) + 1 = \frac{1}{4}(x + 2)
\]
This can be rearranged to give:
\[
f(x) = \frac{1}{4}(x + 2) - 1
\]
Now simplify the function:
\[
f(x) = \frac{1}{4}x + \frac{2}{4} - 1
\]
\[
f(x) = \frac{1}{4}x + \frac{1}{2} - 1
\]
\[
f(x) = \frac{1}{4}x - \frac{1}{2}
\]
### Summary
1. **Equation in Point-Slope Form**: \(y + 1 = \frac{1}{4}(x + 2)\)
2. **Function Notation**: \(f(x) = \frac{1}{4}x - \frac{1}{2}\)
Feel free to ask if you need any further assistance!