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Sagot :
To graph the line [tex]\(y = -2x\)[/tex] and identify specific points on the line, follow these steps:
### 1. Understand the Equation
The equation given is [tex]\( y = -2x \)[/tex]. This is a linear equation in slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Here:
- The slope [tex]\( m \)[/tex] is -2, which means for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2 units.
- The y-intercept [tex]\( b \)[/tex] is 0, meaning the line passes through the origin (0, 0).
### 2. Calculate the Points on the Line
We'll calculate the corresponding [tex]\( y \)[/tex] values for the given [tex]\( x \)[/tex] values of [tex]\(-4\)[/tex] and [tex]\(8\)[/tex].
#### For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -2x \][/tex]
[tex]\[ y = -2(-4) \][/tex]
[tex]\[ y = 8 \][/tex]
So, the point is [tex]\((-4, 8)\)[/tex].
#### For [tex]\( x = 8 \)[/tex]:
[tex]\[ y = -2x \][/tex]
[tex]\[ y = -2(8) \][/tex]
[tex]\[ y = -16 \][/tex]
So, the point is [tex]\((8, -16)\)[/tex].
### 3. Plot the Line and Points
To graph the line [tex]\( y = -2x \)[/tex] and plot the points, follow these steps:
#### Plotting the Line:
1. Choose a Range of [tex]\( x \)[/tex] Values: Typically, you choose values from both negative and positive side. For example, from [tex]\( -10 \)[/tex] to [tex]\( 10 \)[/tex].
2. Calculate Corresponding [tex]\( y \)[/tex] Values: Use the equation [tex]\( y = -2x \)[/tex] to find [tex]\( y \)[/tex] for all chosen [tex]\( x \)[/tex] values.
3. Draw the Line: Connect the points determined by the [tex]\( (x, y) \)[/tex] pairs.
#### Plotting the Points:
1. Identify the points:
- [tex]\((-4, 8)\)[/tex]
- [tex]\((8, -16)\)[/tex]
2. Plot these specific points on the graph.
### 4. Sketch the Graph
1. X and Y Axes: Draw [tex]\( x \)[/tex]-axis (horizontal) and [tex]\( y \)[/tex]-axis (vertical).
2. Plot Selected [tex]\( x \)[/tex] and [tex]\( y \)[/tex] Ranges: Choose a proper range for [tex]\( x \)[/tex] values (e.g., from [tex]\([-10, 10]\)[/tex]) and calculate the corresponding [tex]\( y \)[/tex] values, but for simplicity, we'll just use these key points.
3. Plot the Line: Start at the origin since the y-intercept [tex]\( b = 0 \)[/tex], and use the slope [tex]\( m = -2 \)[/tex] to go down 2 units for each unit to the right.
4. Highlight the Points:
- [tex]\((-4, 8)\)[/tex]: Plot this point by going left 4 units on the [tex]\( x \)[/tex]-axis and up 8 units on the [tex]\( y \)[/tex]-axis.
- [tex]\((8, -16)\)[/tex]: Plot this point by going right 8 units on the [tex]\( x \)[/tex]-axis and down 16 units on the [tex]\( y \)[/tex]-axis.
### 5. Label the Graph
- Axes Labels: Label the [tex]\( x \)[/tex]-axis and [tex]\( y \)[/tex]-axis.
- Title: Add a title to the graph such as "Graph of [tex]\( y = -2x \)[/tex]".
- Point Labels: Label the points [tex]\((-4, 8)\)[/tex] and [tex]\((8, -16)\)[/tex] clearly on the graph.
### Summary
- [tex]\( y = -2x \)[/tex] produces a straight line with a slope of -2, starting from the origin.
- Key points calculated are [tex]\((-4, 8)\)[/tex] and [tex]\((8, -16)\)[/tex].
- Plot these points and draw the line through them for visual confirmation of the equation on a graph.
By following these steps, you should be able to graph the line and understand the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation [tex]\( y = -2x \)[/tex].
### 1. Understand the Equation
The equation given is [tex]\( y = -2x \)[/tex]. This is a linear equation in slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Here:
- The slope [tex]\( m \)[/tex] is -2, which means for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2 units.
- The y-intercept [tex]\( b \)[/tex] is 0, meaning the line passes through the origin (0, 0).
### 2. Calculate the Points on the Line
We'll calculate the corresponding [tex]\( y \)[/tex] values for the given [tex]\( x \)[/tex] values of [tex]\(-4\)[/tex] and [tex]\(8\)[/tex].
#### For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -2x \][/tex]
[tex]\[ y = -2(-4) \][/tex]
[tex]\[ y = 8 \][/tex]
So, the point is [tex]\((-4, 8)\)[/tex].
#### For [tex]\( x = 8 \)[/tex]:
[tex]\[ y = -2x \][/tex]
[tex]\[ y = -2(8) \][/tex]
[tex]\[ y = -16 \][/tex]
So, the point is [tex]\((8, -16)\)[/tex].
### 3. Plot the Line and Points
To graph the line [tex]\( y = -2x \)[/tex] and plot the points, follow these steps:
#### Plotting the Line:
1. Choose a Range of [tex]\( x \)[/tex] Values: Typically, you choose values from both negative and positive side. For example, from [tex]\( -10 \)[/tex] to [tex]\( 10 \)[/tex].
2. Calculate Corresponding [tex]\( y \)[/tex] Values: Use the equation [tex]\( y = -2x \)[/tex] to find [tex]\( y \)[/tex] for all chosen [tex]\( x \)[/tex] values.
3. Draw the Line: Connect the points determined by the [tex]\( (x, y) \)[/tex] pairs.
#### Plotting the Points:
1. Identify the points:
- [tex]\((-4, 8)\)[/tex]
- [tex]\((8, -16)\)[/tex]
2. Plot these specific points on the graph.
### 4. Sketch the Graph
1. X and Y Axes: Draw [tex]\( x \)[/tex]-axis (horizontal) and [tex]\( y \)[/tex]-axis (vertical).
2. Plot Selected [tex]\( x \)[/tex] and [tex]\( y \)[/tex] Ranges: Choose a proper range for [tex]\( x \)[/tex] values (e.g., from [tex]\([-10, 10]\)[/tex]) and calculate the corresponding [tex]\( y \)[/tex] values, but for simplicity, we'll just use these key points.
3. Plot the Line: Start at the origin since the y-intercept [tex]\( b = 0 \)[/tex], and use the slope [tex]\( m = -2 \)[/tex] to go down 2 units for each unit to the right.
4. Highlight the Points:
- [tex]\((-4, 8)\)[/tex]: Plot this point by going left 4 units on the [tex]\( x \)[/tex]-axis and up 8 units on the [tex]\( y \)[/tex]-axis.
- [tex]\((8, -16)\)[/tex]: Plot this point by going right 8 units on the [tex]\( x \)[/tex]-axis and down 16 units on the [tex]\( y \)[/tex]-axis.
### 5. Label the Graph
- Axes Labels: Label the [tex]\( x \)[/tex]-axis and [tex]\( y \)[/tex]-axis.
- Title: Add a title to the graph such as "Graph of [tex]\( y = -2x \)[/tex]".
- Point Labels: Label the points [tex]\((-4, 8)\)[/tex] and [tex]\((8, -16)\)[/tex] clearly on the graph.
### Summary
- [tex]\( y = -2x \)[/tex] produces a straight line with a slope of -2, starting from the origin.
- Key points calculated are [tex]\((-4, 8)\)[/tex] and [tex]\((8, -16)\)[/tex].
- Plot these points and draw the line through them for visual confirmation of the equation on a graph.
By following these steps, you should be able to graph the line and understand the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation [tex]\( y = -2x \)[/tex].
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