Question
A coordinate plane with a line passing through (negative 3, 2) and (0, 3)Find the linear function represented by the graph.The slope of the line is .The y-intercept of the line is at .What linear function is represented by the graph?
Answers
Answer
Find the linear function represented by the graph.
The slope of the line is
✔ 1/3
.
The y-intercept of the line is at
✔ 3
.
What linear function is represented by the graph?
✔ f(x) = (1/3)x + 3
Write the point-slope form of the equation for a line that passes through (6, –1) with a slope of 2.
The value of x1 is
.
The value of y1 is
.
The point-slope form of the equation is
.
The slope of the line is
✔ 1/3
.
The y-intercept of the line is at
✔ 3
.
What linear function is represented by the graph?
✔ f(x) = (1/3)x + 3
Write the point-slope form of the equation for a line that passes through (6, –1) with a slope of 2.
The value of x1 is
.
The value of y1 is
.
The point-slope form of the equation is
.
Answer
Write the point-slope form of the equation for a line that passes through (6, –1) with a slope of 2.
The value of x1 is
✔ 6
.
The value of y1 is
✔ –1
.
The point-slope form of the equation is
✔ y + 1 = 2(x – 6)
.
A coordinate plane with a line passing through points (1, 2) and (4, 4)The slope of the graphed line is . Which formulas represent the line that is graphed? Check all that apply.y – 1 = 2/3 (x – 2)y – 2 = 2/3 (x – 1)y – 4 = 2/3 (x – 4)f(x) = 2/3 x +1/3 f(x) =2/3 x +4/3
The value of x1 is
✔ 6
.
The value of y1 is
✔ –1
.
The point-slope form of the equation is
✔ y + 1 = 2(x – 6)
.
A coordinate plane with a line passing through points (1, 2) and (4, 4)The slope of the graphed line is . Which formulas represent the line that is graphed? Check all that apply.y – 1 = 2/3 (x – 2)y – 2 = 2/3 (x – 1)y – 4 = 2/3 (x – 4)f(x) = 2/3 x +1/3 f(x) =2/3 x +4/3
Answer
A tub of water is emptied at a rate of 3 gallons per minute. The equation y –12 = –3(x – 1) models the amount of water remaining, where x is time (in minutes) and y is the amount of water left (in gallons). Analyze the work shown below to determine the initial amount of water.
1. Solve for the y-variable.
y – 12 = –3(x – 1)
y – 12 = –3x + 3
y = –3x +15
2. Write the equation using function notation.
f(x) = –3x +15
The tub started with________
gallons of water.
1. Solve for the y-variable.
y – 12 = –3(x – 1)
y – 12 = –3x + 3
y = –3x +15
2. Write the equation using function notation.
f(x) = –3x +15
The tub started with________
gallons of water.
Answered by
GPT-4o mini
To find the linear function represented by the line passing through the points \((-3, 2)\) and \((0, 3)\), we first need to calculate the slope of the line and then use the slope-intercept form of a linear equation.
### Step 1: Calculate the slope
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-3, 2)\) (let's call this \((x_1, y_1)\)) and \((0, 3)\) (let's call this \((x_2, y_2)\)):
\[
m = \frac{3 - 2}{0 - (-3)} = \frac{1}{3}
\]
So, the slope of the line is \(\frac{1}{3}\).
### Step 2: Identify the y-intercept
The y-intercept is the value of \(y\) when \(x = 0\). From the given point \((0, 3)\), we see that the y-intercept is \(3\).
### Step 3: Write the linear function
The slope-intercept form of a linear equation is given by:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept. Plugging in the values we found:
\[
y = \frac{1}{3}x + 3
\]
### Summary:
- The slope of the line is \(\frac{1}{3}\).
- The y-intercept of the line is \(3\).
- The linear function represented by the graph is:
\[
y = \frac{1}{3}x + 3
\]
### Step 1: Calculate the slope
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-3, 2)\) (let's call this \((x_1, y_1)\)) and \((0, 3)\) (let's call this \((x_2, y_2)\)):
\[
m = \frac{3 - 2}{0 - (-3)} = \frac{1}{3}
\]
So, the slope of the line is \(\frac{1}{3}\).
### Step 2: Identify the y-intercept
The y-intercept is the value of \(y\) when \(x = 0\). From the given point \((0, 3)\), we see that the y-intercept is \(3\).
### Step 3: Write the linear function
The slope-intercept form of a linear equation is given by:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept. Plugging in the values we found:
\[
y = \frac{1}{3}x + 3
\]
### Summary:
- The slope of the line is \(\frac{1}{3}\).
- The y-intercept of the line is \(3\).
- The linear function represented by the graph is:
\[
y = \frac{1}{3}x + 3
\]
Answered by
GPT-4o mini
To write the point-slope form of the equation for a line that passes through the point \((6, -1)\) with a slope of \(2\), we will use the point-slope form formula:
\[
y - y_1 = m(x - x_1)
\]
where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line.
### Given:
- Point: \((x_1, y_1) = (6, -1)\)
- Slope: \(m = 2\)
### Step 1: Identify \(x_1\) and \(y_1\)
From the point \((6, -1)\):
- **The value of \(x_1\)** is \(6\).
- **The value of \(y_1\)** is \(-1\).
### Step 2: Write the point-slope form
Now, substituting \(x_1\), \(y_1\), and \(m\) into the point-slope formula:
\[
y - (-1) = 2(x - 6)
\]
This simplifies to:
\[
y + 1 = 2(x - 6)
\]
### Summary:
- The value of \(x_1\) is **6**.
- The value of \(y_1\) is **-1**.
- The point-slope form of the equation is:
\[
y + 1 = 2(x - 6)
\]
\[
y - y_1 = m(x - x_1)
\]
where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line.
### Given:
- Point: \((x_1, y_1) = (6, -1)\)
- Slope: \(m = 2\)
### Step 1: Identify \(x_1\) and \(y_1\)
From the point \((6, -1)\):
- **The value of \(x_1\)** is \(6\).
- **The value of \(y_1\)** is \(-1\).
### Step 2: Write the point-slope form
Now, substituting \(x_1\), \(y_1\), and \(m\) into the point-slope formula:
\[
y - (-1) = 2(x - 6)
\]
This simplifies to:
\[
y + 1 = 2(x - 6)
\]
### Summary:
- The value of \(x_1\) is **6**.
- The value of \(y_1\) is **-1**.
- The point-slope form of the equation is:
\[
y + 1 = 2(x - 6)
\]
Answered by
GPT-4o mini
### Step 1: Calculate the Slope
To find the slope \(m\) of the line that passes through the points \((1, 2)\) and \((4, 4)\), we use the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting in the coordinates:
\[
m = \frac{4 - 2}{4 - 1} = \frac{2}{3}
\]
So, the slope of the line is \(\frac{2}{3}\).
### Step 2: Write the Point-Slope Form
Using one of the points, say \((1, 2)\), we can write the point-slope form of the equation:
\[
y - y_1 = m(x - x_1)
\]
Substituting \(m = \frac{2}{3}\), \(x_1 = 1\), and \(y_1 = 2\):
\[
y - 2 = \frac{2}{3}(x - 1)
\]
### Step 3: Check the Given Equations
Now let's check which of the formulas represent the line that is graphed:
1. **\(\mathbf{y - 1 = \frac{2}{3} (x - 2)}\)**:
- This is not valid since it does not pass through either of the points we have.
2. **\(\mathbf{y - 2 = \frac{2}{3} (x - 1)}\)**:
- This is the point-slope form we derived, so it is valid.
3. **\(\mathbf{y - 4 = \frac{2}{3} (x - 4)}\)**:
- Testing the point \((4, 4)\):
- \(y - 4 = \frac{2}{3}(4 - 4) \Rightarrow y - 4 = 0\) ⇒ \(y = 4\). \(\text{Valid}\)
4. **\(\mathbf{f(x) = \frac{2}{3} x + \frac{1}{3}}\)**:
- To convert this into slope-intercept form, set \(y = \frac{2}{3}x + \frac{1}{3}\).
- If we substitute \(x = 1\), \(y = \frac{2}{3}(1) + \frac{1}{3} = 1\), so it does not go through both points.
- Not valid.
5. **\(\mathbf{f(x) = \frac{2}{3} x + \frac{4}{3}}\)**:
- Converting to slope-intercept form, \(y = \frac{2}{3}x + \frac{4}{3}\).
- For \(x = 1\), \(y = \frac{2}{3}(1) + \frac{4}{3} = \frac{6}{3} = 2\) (Valid for point (1, 2))
- For \(x = 4\), \(y = \frac{2}{3}(4) + \frac{4}{3} = \frac{8}{3} + \frac{4}{3} = 4\) (Valid for point (4, 4))
- So, this is valid.
### Summary of Valid Equations:
- **Valid equations are:**
- \(\mathbf{y - 2 = \frac{2}{3} (x - 1)}\)
- \(\mathbf{y - 4 = \frac{2}{3} (x - 4)}\)
- \(\mathbf{f(x) = \frac{2}{3} x + \frac{4}{3}}\)
### Conclusion:
The valid formulas that represent the line graphed are:
- \(y - 2 = \frac{2}{3} (x - 1)\)
- \(y - 4 = \frac{2}{3} (x - 4)\)
- \(f(x) = \frac{2}{3} x + \frac{4}{3}\)
To find the slope \(m\) of the line that passes through the points \((1, 2)\) and \((4, 4)\), we use the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting in the coordinates:
\[
m = \frac{4 - 2}{4 - 1} = \frac{2}{3}
\]
So, the slope of the line is \(\frac{2}{3}\).
### Step 2: Write the Point-Slope Form
Using one of the points, say \((1, 2)\), we can write the point-slope form of the equation:
\[
y - y_1 = m(x - x_1)
\]
Substituting \(m = \frac{2}{3}\), \(x_1 = 1\), and \(y_1 = 2\):
\[
y - 2 = \frac{2}{3}(x - 1)
\]
### Step 3: Check the Given Equations
Now let's check which of the formulas represent the line that is graphed:
1. **\(\mathbf{y - 1 = \frac{2}{3} (x - 2)}\)**:
- This is not valid since it does not pass through either of the points we have.
2. **\(\mathbf{y - 2 = \frac{2}{3} (x - 1)}\)**:
- This is the point-slope form we derived, so it is valid.
3. **\(\mathbf{y - 4 = \frac{2}{3} (x - 4)}\)**:
- Testing the point \((4, 4)\):
- \(y - 4 = \frac{2}{3}(4 - 4) \Rightarrow y - 4 = 0\) ⇒ \(y = 4\). \(\text{Valid}\)
4. **\(\mathbf{f(x) = \frac{2}{3} x + \frac{1}{3}}\)**:
- To convert this into slope-intercept form, set \(y = \frac{2}{3}x + \frac{1}{3}\).
- If we substitute \(x = 1\), \(y = \frac{2}{3}(1) + \frac{1}{3} = 1\), so it does not go through both points.
- Not valid.
5. **\(\mathbf{f(x) = \frac{2}{3} x + \frac{4}{3}}\)**:
- Converting to slope-intercept form, \(y = \frac{2}{3}x + \frac{4}{3}\).
- For \(x = 1\), \(y = \frac{2}{3}(1) + \frac{4}{3} = \frac{6}{3} = 2\) (Valid for point (1, 2))
- For \(x = 4\), \(y = \frac{2}{3}(4) + \frac{4}{3} = \frac{8}{3} + \frac{4}{3} = 4\) (Valid for point (4, 4))
- So, this is valid.
### Summary of Valid Equations:
- **Valid equations are:**
- \(\mathbf{y - 2 = \frac{2}{3} (x - 1)}\)
- \(\mathbf{y - 4 = \frac{2}{3} (x - 4)}\)
- \(\mathbf{f(x) = \frac{2}{3} x + \frac{4}{3}}\)
### Conclusion:
The valid formulas that represent the line graphed are:
- \(y - 2 = \frac{2}{3} (x - 1)\)
- \(y - 4 = \frac{2}{3} (x - 4)\)
- \(f(x) = \frac{2}{3} x + \frac{4}{3}\)
Answered by
GPT-4o mini
To analyze the equation that models the amount of water remaining, let's take a closer look at the equation provided:
### Given Equation
The equation is:
\[
y - 12 = -3(x - 1)
\]
### Step 1: Solve for \(y\)
You have already solved for \(y\) correctly. Let's recap that work:
1. Start with the equation:
\[
y - 12 = -3(x - 1)
\]
2. Distributing \(-3\):
\[
y - 12 = -3x + 3
\]
3. Adding \(12\) to both sides:
\[
y = -3x + 15
\]
### Step 2: Write in Function Notation
You correctly converted this to function notation:
\[
f(x) = -3x + 15
\]
### Step 3: Determine the Initial Amount of Water
To find the initial amount of water in the tub, we need to evaluate the function when \(x = 0\) (the start time).
Substituting \(x = 0\) into the equation \(f(x)\):
\[
f(0) = -3(0) + 15 = 15
\]
### Conclusion
The tub started with **15 gallons** of water.
### Given Equation
The equation is:
\[
y - 12 = -3(x - 1)
\]
### Step 1: Solve for \(y\)
You have already solved for \(y\) correctly. Let's recap that work:
1. Start with the equation:
\[
y - 12 = -3(x - 1)
\]
2. Distributing \(-3\):
\[
y - 12 = -3x + 3
\]
3. Adding \(12\) to both sides:
\[
y = -3x + 15
\]
### Step 2: Write in Function Notation
You correctly converted this to function notation:
\[
f(x) = -3x + 15
\]
### Step 3: Determine the Initial Amount of Water
To find the initial amount of water in the tub, we need to evaluate the function when \(x = 0\) (the start time).
Substituting \(x = 0\) into the equation \(f(x)\):
\[
f(0) = -3(0) + 15 = 15
\]
### Conclusion
The tub started with **15 gallons** of water.