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Graph y and x.

y = 1/4x + 2

11 answers

To graph the linear equation \( y = \frac{1}{4}x + 2 \), you can follow these steps:

  1. Identify the Slope and Y-intercept:

    • The equation is in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
    • From the equation \( y = \frac{1}{4}x + 2 \), the slope \( m \) is \( \frac{1}{4} \) and the y-intercept \( b \) is 2.

  2. Plot the Y-intercept:

    • Start by plotting the point where the line crosses the y-axis, which is at (0, 2).

  3. Use the Slope to Find Another Point:

    • The slope \( \frac{1}{4} \) means that for every 4 units you move to the right (positive x-direction), you move up 1 unit (positive y-direction).
    • From the y-intercept (0, 2), move 4 units to the right to x = 4 and 1 unit up to y = 3. This gives you the point (4, 3).

  4. Plot the Second Point:

    • Plot the point (4, 3) on the graph.

  5. Draw the Line:

    • Draw a straight line through the points (0, 2) and (4, 3). Extend the line in both directions.

  6. Check Additional Points (Optional):

    • You can find more points by choosing other values for \( x \):
      • If \( x = -4 \), then \( y = \frac{1}{4}(-4) + 2 = 1 \), which gives the point (-4, 1).
      • If \( x = 4 \), then \( y = 3 \) (already plotted).
      • If \( x = 8 \), then \( y = \frac{1}{4}(8) + 2 = 4 \), which gives the point (8, 4).

Now you have the graph of the equation \( y = \frac{1}{4}x + 2 \) with key points and a straight line.

If you need a visual, you can use graphing tools or software to plot it based on these instructions!

Graph y and x.
y = 0.75x + 5

To graph the linear equation \( y = 0.75x + 5 \), you can follow these steps:

Step 1: Identify the Slope and Y-intercept

  • The equation is in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  • Here, the slope \( m \) is \( 0.75 \) and the y-intercept \( b \) is \( 5 \).

Step 2: Plot the Y-intercept

  • Start by plotting the y-intercept (0, 5) on the graph. This is where the line crosses the y-axis.

Step 3: Use the Slope to Find Another Point

  • The slope \( 0.75 \) can be expressed as \( \frac{3}{4} \), which means that for every 4 units you move to the right (positive x-direction), you move up 3 units (positive y-direction).
  • Starting from the point (0, 5):
    • Move 4 units to the right to \( x = 4 \).
    • Move up 3 units to \( y = 8 \).
  • This gives you another point: (4, 8).

Step 4: Plot the Second Point

  • Plot the point (4, 8) on the graph.

Step 5: Draw the Line

  • Draw a straight line through the points (0, 5) and (4, 8). Extend the line in both directions.

Step 6: Check Additional Points (Optional)

To ensure accuracy, you can calculate more points:

  • If \( x = -4 \): \[ y = 0.75(-4) + 5 = -3 + 5 = 2 \quad \text{(Point: (-4, 2))} \]
  • If \( x = 8 \): \[ y = 0.75(8) + 5 = 6 + 5 = 11 \quad \text{(Point: (8, 11))} \]

Final Graph

  • The line will pass through the points (0, 5), (4, 8), (-4, 2), and (8, 11).
  • Make sure to draw an arrow on both ends of the line indicating it extends infinitely.

If you are using graph paper or a graphing tool, you can plot these points and draw the line accordingly to visualize the equation \( y = 0.75x + 5 \).

Graph y and x.
y + 1 = 3(x + 6)

To graph the equation \( y + 1 = 3(x + 6) \), we need to rearrange it into slope-intercept form \( y = mx + b \).

Step 1: Simplify the Equation

Starting with: \[ y + 1 = 3(x + 6) \]

Distributing the \( 3 \): \[ y + 1 = 3x + 18 \]

Next, isolate \( y \) by subtracting \( 1 \) from both sides: \[ y = 3x + 18 - 1 \] \[ y = 3x + 17 \]

Step 2: Identify the Slope and Y-intercept

Now, we have the equation in slope-intercept form \( y = mx + b \):

  • The slope \( m \) is \( 3 \).
  • The y-intercept \( b \) is \( 17 \).

Step 3: Plot the Y-intercept

The y-intercept is the point where the line crosses the y-axis:

  • Plot the point \( (0, 17) \).

Step 4: Use the Slope to Find Another Point

The slope \( 3 \) can be written as \( \frac{3}{1} \), which means:

  • For every \( 1 \) unit you move right (in the positive x-direction), you move up \( 3 \) units.

Starting from the point \( (0, 17) \):

  • Move \( 1 \) unit to the right to \( x = 1 \).
  • Move \( 3 \) units up to \( y = 20 \).
  • This gives you another point: \( (1, 20) \).

Step 5: Plot the Second Point

  • Plot the point \( (1, 20) \) on the graph.

Step 6: Draw the Line

  • Draw a straight line through the points \( (0, 17) \) and \( (1, 20) \). Extend the line in both directions.

Step 7: Check Additional Points (Optional)

To ensure accuracy, you can calculate more points:

  • If \( x = -6 \): \[ y = 3(-6) + 17 = -18 + 17 = -1 \quad \text{(Point: (-6, -1))} \]
  • If \( x = 2 \): \[ y = 3(2) + 17 = 6 + 17 = 23 \quad \text{(Point: (2, 23))} \]

Final Graph

  • The line will pass through the points \( (0, 17) \), \( (1, 20) \), \( (-6, -1) \), and \( (2, 23) \).
  • Make sure to draw an arrow on both ends of the line indicating it extends infinitely.

If you are using graph paper or a graphing tool, you can plot these points and draw the line accordingly to visualize the equation \( y + 1 = 3(x + 6) \).

Graph y and x.
y − 6 = −34(x − 2)

To graph the equation \( y - 6 = -34(x - 2) \), we will first rearrange it into slope-intercept form \( y = mx + b \).

Step 1: Simplify the Equation

Starting with: \[ y - 6 = -34(x - 2) \]

Distributing \(-34\): \[ y - 6 = -34x + 68 \]

Now, isolate \(y\) by adding \(6\) to both sides: \[ y = -34x + 68 + 6 \] \[ y = -34x + 74 \]

Step 2: Identify the Slope and Y-intercept

Now the equation is in slope-intercept form \( y = mx + b \):

  • The slope \( m \) is \(-34\).
  • The y-intercept \( b \) is \( 74 \).

Step 3: Plot the Y-intercept

The y-intercept is the point where the line crosses the y-axis:

  • Plot the point \( (0, 74) \).

Step 4: Use the Slope to Find Another Point

The slope \(-34\) means for every \( 1 \) unit you move to the right (positive x-direction), you move down \( 34 \) units (because the slope is negative).

Starting from the point \( (0, 74) \):

  • Move \( 1 \) unit to the right to \( x = 1 \).
  • Move \( 34 \) units down to \( y = 74 - 34 = 40 \).
  • This gives you another point: \( (1, 40) \).

Step 5: Plot the Second Point

  • Plot the point \( (1, 40) \) on the graph.

Step 6: Draw the Line

  • Draw a straight line through the points \( (0, 74) \) and \( (1, 40) \). Extend the line in both directions.

Step 7: Check Additional Points (Optional)

To confirm accuracy, you can calculate more points:

  • If \( x = 2 \): \[ y = -34(2) + 74 = -68 + 74 = 6 \quad \text{(Point: (2, 6))} \]
  • If \( x = -1 \): \[ y = -34(-1) + 74 = 34 + 74 = 108 \quad \text{(Point: (-1, 108))} \]

Final Graph

  • The line will pass through the points \( (0, 74) \), \( (1, 40) \), \( (2, 6) \), and \( (-1, 108) \).
  • Make sure to draw an arrow on both ends of the line indicating it extends infinitely.

If you're using graph paper or a graphing tool, you can plot these points and draw the line accordingly to visualize the equation \( y - 6 = -34(x - 2) \).

The graph is 1 to 14 and -1 to -14.

To graph the equation \( y - 6 = -34(x - 2) \) correctly on a coordinate plane that spans from \( x = -1 \) to \( x = 14 \) and from \( y = -1 \) to \( y = 14 \), we need to follow the previous steps and refine the plotting based on the specified range.

Step 1: Simplify the Equation

We already simplified the equation to:

\[ y = -34x + 74 \]

Step 2: Identify the Y-intercept

  • The y-intercept is \( (0, 74) \). However, since the y-values are limited to \( -1 \) to \( 14 \), this point lies outside the given graph range.

Step 3: Find Plotting Points Within the Range

We need some points within the specified range:

  • Choose \( x = 1 \): \[ y = -34(1) + 74 = -34 + 74 = 40 \quad \text{(Point: (1, 40))} \]

  • Choose \( x = 2 \): \[ y = -34(2) + 74 = -68 + 74 = 6 \quad \text{(Point: (2, 6))} \]

  • Choose \( x = 3 \): \[ y = -34(3) + 74 = -102 + 74 = -28 \quad \text{(Point: (3, -28) - outside range)} \]

Since \( x = 1 \) and \( x = 2 \) give us one point that lies within the range, we should find another lower \( x \) to plot:

  • Choose \( x = 13 \): \[ y = -34(13) + 74 = -442 + 74 = -368 \quad \text{(Point: (13, -368) - outside range)} \]

  • Choose \( x = -1 \): \[ y = -34(-1) + 74 = 34 + 74 = 108 \quad \text{(Point: (-1, 108) - outside range)} \]

To find practical points that fit within the limits from \( y = -1 \) to \( y = 14 \):

  • Calculate if \( y = 14 \): \[ 14 = -34x + 74 \quad \Rightarrow \quad 34x = 74 - 14 \] \[ 34x = 60 \quad \Rightarrow \quad x = \frac{60}{34} \approx 1.76 \quad \text{(Point: (1.76, 14))} \]

Summary of Points Within the Range

  1. (2, 6) - Valid point.
  2. (1, 40) - Not valid for graphs ranging from -1 to 14.
  3. (1.76, 14) - Valid point.

Drawing the Line

With the valid points \( (2, 6) \) and \( (1.76, 14) \):

  • You can plot these on your graph.
  • (2, 6) is located near the center of the \( 1 \) to \( 14 \) height.
  • (1.76, 14) is near the top of the graph where the y-value equals \( 14 \).

Final Graph Instructions

  1. Plot the points \( (2, 6) \), and \( (1.76, 14) \).
  2. Because the line has a steep negative slope, the trend would be that the line decreases rapidly across the \( x \)-axis.
  3. If extending to other quadrants, remember that the line continues downward outside of your box, and adjust for visibility.

Make sure to draw arrows on both ends of the line, indicating that the line continues a pattern.

If you have graphing software or graph paper, use these points for plotting to visualize how the line behaves within the given constraints.

Graph y and x.
4y = 3x − 16

To graph the equation \(4y = 3x - 16\), we'll first convert it to slope-intercept form \(y = mx + b\), identify key points, and then graph the equation.

Step 1: Convert to Slope-Intercept Form

Starting with the equation: \[ 4y = 3x - 16 \]

Now, divide each term by 4 to solve for \(y\): \[ y = \frac{3}{4}x - 4 \]

Step 2: Identify the Slope and Y-intercept

Now that we have the equation in slope-intercept form \(y = mx + b\):

  • The slope \(m\) is \(\frac{3}{4}\).
  • The y-intercept \(b\) is \(-4\).

Step 3: Plot the Y-intercept

The y-intercept is the point where the line crosses the y-axis:

  • Plot the point \((0, -4)\).

Step 4: Use the Slope to Find Another Point

The slope \(\frac{3}{4}\) means for every \(4\) units you move to the right (positive x-direction), you move \(3\) units up (positive y-direction).

Starting from the y-intercept \((0, -4)\):

  1. Move \(4\) units to the right to \(x = 4\).
  2. Move \(3\) units up to \(y = -4 + 3 = -1\).
  • This gives you another point: \((4, -1)\).

Step 5: Draw the Line

  • Plot the points \((0, -4)\) and \((4, -1)\).
  • Draw a straight line through the points and extend it in both directions.

Step 6: Check Additional Points (Optional)

To ensure accuracy, you can find more points by choosing different \(x\) values:

  • If \(x = 8\): \[ y = \frac{3}{4}(8) - 4 = 6 - 4 = 2 \quad \text{(Point: (8, 2))} \]

  • If \(x = -4\): \[ y = \frac{3}{4}(-4) - 4 = -3 - 4 = -7 \quad \text{(Point: (-4, -7))} \]

Final Points to Plot

  • We can now plot:
    • The y-intercept \((0, -4)\)
    • The point \((4, -1)\)
    • The point \((8, 2)\)

Summary

  1. Plot the points: \((0, -4)\), \((4, -1)\), and \((8, 2)\).
  2. Draw a line through these points, extending in both directions.

This will give you the graph of the equation \(4y = 3x - 16\). If you're working with graphing tools or software, these points are sufficient to visualize the line as well!