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?

Answer 1

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:

\[ y = \frac{1}{3}x + 3 \]

Answer 2

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
.

Answer 3

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:

\[ y + 1 = 2(x - 6) \]

Answer 4

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

Answer 5

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:

• \(y - 2 = \frac{2}{3} (x - 1)\)
• \(y - 4 = \frac{2}{3} (x - 4)\)
• \(f(x) = \frac{2}{3} x + \frac{4}{3}\)

Answer 6

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.

Answer 7

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.