Answer :
Certainly! Let’s break down the problem step-by-step:
### Step 1: Determine the Coordinates of the Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. At this point, the value of [tex]\(x\)[/tex] is 0.
Given the equation of the line:
[tex]\[ y = 2x - 1 \][/tex]
To find the y-intercept:
- Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = 2(0) - 1 \][/tex]
[tex]\[ y = -1 \][/tex]
So, the coordinates of the y-intercept are:
[tex]\[ (0, -1) \][/tex]
### Step 2: Determine the Coordinates of the X-Intercept
The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of [tex]\(y\)[/tex] is 0.
Given the equation of the line:
[tex]\[ y = 2x - 1 \][/tex]
To find the x-intercept:
- Set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = 2x - 1 \][/tex]
[tex]\[ 1 = 2x \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
So, the coordinates of the x-intercept are:
[tex]\[ \left( \frac{1}{2}, 0 \right) \][/tex]
### Step 3: Draw the Graph of the Function [tex]\( y = 2x - 1 \)[/tex]
To draw the graph of the function, you can follow these steps:
1. Plot the intercepts:
- Plot the y-intercept at [tex]\((0, -1)\)[/tex].
- Plot the x-intercept at [tex]\(\left( \frac{1}{2}, 0 \right)\)[/tex].
2. Draw the line:
- Since the equation describes a straight line, you only need two points to draw it. Use the intercepts to guide your line.
- Draw a line passing through the points [tex]\((0, -1)\)[/tex] and [tex]\(\left( \frac{1}{2}, 0 \right)\)[/tex]. Extend the line in both directions to cover a reasonable range of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values.
To visualize this, the graph of [tex]\( y = 2x - 1 \)[/tex] will have an upward slope, because the coefficient of [tex]\( x \)[/tex] (which is 2) is positive. The y-intercept is at [tex]\((0, -1)\)[/tex], meaning the line crosses the y-axis below the origin. The x-intercept is at [tex]\(\left( \frac{1}{2}, 0 \right)\)[/tex], meaning the line crosses the x-axis to the right of the origin.
Here's a rough sketch of what the graph would look like:
```
y
^
| /
| /
| /
| /
| /
| /
| /
| /
|
---------------------------------> x
-1 0 1/2
(0, -1) (0,0) (1/2,0)
```
The line passes through the points [tex]\((0, -1)\)[/tex] and [tex]\(\left( \frac{1}{2}, 0 \right)\)[/tex], clearly showing the intercepts on the graph.
### Step 1: Determine the Coordinates of the Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. At this point, the value of [tex]\(x\)[/tex] is 0.
Given the equation of the line:
[tex]\[ y = 2x - 1 \][/tex]
To find the y-intercept:
- Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = 2(0) - 1 \][/tex]
[tex]\[ y = -1 \][/tex]
So, the coordinates of the y-intercept are:
[tex]\[ (0, -1) \][/tex]
### Step 2: Determine the Coordinates of the X-Intercept
The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of [tex]\(y\)[/tex] is 0.
Given the equation of the line:
[tex]\[ y = 2x - 1 \][/tex]
To find the x-intercept:
- Set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = 2x - 1 \][/tex]
[tex]\[ 1 = 2x \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
So, the coordinates of the x-intercept are:
[tex]\[ \left( \frac{1}{2}, 0 \right) \][/tex]
### Step 3: Draw the Graph of the Function [tex]\( y = 2x - 1 \)[/tex]
To draw the graph of the function, you can follow these steps:
1. Plot the intercepts:
- Plot the y-intercept at [tex]\((0, -1)\)[/tex].
- Plot the x-intercept at [tex]\(\left( \frac{1}{2}, 0 \right)\)[/tex].
2. Draw the line:
- Since the equation describes a straight line, you only need two points to draw it. Use the intercepts to guide your line.
- Draw a line passing through the points [tex]\((0, -1)\)[/tex] and [tex]\(\left( \frac{1}{2}, 0 \right)\)[/tex]. Extend the line in both directions to cover a reasonable range of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values.
To visualize this, the graph of [tex]\( y = 2x - 1 \)[/tex] will have an upward slope, because the coefficient of [tex]\( x \)[/tex] (which is 2) is positive. The y-intercept is at [tex]\((0, -1)\)[/tex], meaning the line crosses the y-axis below the origin. The x-intercept is at [tex]\(\left( \frac{1}{2}, 0 \right)\)[/tex], meaning the line crosses the x-axis to the right of the origin.
Here's a rough sketch of what the graph would look like:
```
y
^
| /
| /
| /
| /
| /
| /
| /
| /
|
---------------------------------> x
-1 0 1/2
(0, -1) (0,0) (1/2,0)
```
The line passes through the points [tex]\((0, -1)\)[/tex] and [tex]\(\left( \frac{1}{2}, 0 \right)\)[/tex], clearly showing the intercepts on the graph.
