graph y=2x+3 and its inverse.
line goes through x = 0 and y = 3 ,
or (0,3)
line goes through x = 5 and y = 13 ,
or (5,13)
now inverse
x=2y+3
or
2 y = x - 3
y = (x/2) - 1.5
when x = 0, y = -1.5
when x = 6, y = +1.5
To graph the equation y = 2x + 3, and its inverse, let's follow these steps:
Step 1: Create a table of values for the equation y = 2x + 3.
Let's select some x-values and find the corresponding y-values using the equation.
For example, let's choose x = -2, -1, 0, 1, and 2.
For x = -2:
y = 2(-2) + 3 = -4 + 3 = -1
For x = -1:
y = 2(-1) + 3 = -2 + 3 = 1
For x = 0:
y = 2(0) + 3 = 0 + 3 = 3
For x = 1:
y = 2(1) + 3 = 2 + 3 = 5
For x = 2:
y = 2(2) + 3 = 4 + 3 = 7
So, the table of values for y = 2x + 3 is:
_________________
| x | y |
|_____|_____|
| -2 | -1 |
| -1 | 1 |
| 0 | 3 |
| 1 | 5 |
| 2 | 7 |
¯¯¯¯¯¯¯¯¯
Step 2: Plot the points from the table on a coordinate plane.
Use the x-values as the x-coordinates and the y-values as the y-coordinates.
Connect the plotted points to form a line.
Step 3: Find the inverse of the equation y = 2x + 3.
To find the inverse, exchange the x and y variables and solve for y.
Swap x and y in the equation: x = 2y + 3
Rearrange the equation to solve for y:
x - 3 = 2y
(x - 3) / 2 = y
So, the inverse equation is y = (x - 3) / 2.
Step 4: Create a table of values for the inverse equation.
Using the same x-values from before, find the corresponding y-values for the inverse equation:
For x = -2:
y = (-2 - 3) / 2 = -5/2
For x = -1:
y = (-1 - 3) / 2 = -4/2 = -2
For x = 0:
y = (0 - 3) / 2 = -3/2
For x = 1:
y = (1 - 3) / 2 = -2/2 = -1
For x = 2:
y = (2 - 3) / 2 = -1/2
So, the table of values for the inverse equation y = (x - 3) / 2 is:
__________________
| x | y |
|_____|______|
| -2 | -5/2 |
| -1 | -2 |
| 0 | -3/2 |
| 1 | -1 |
| 2 | -1/2 |
¯¯¯¯¯¯¯¯¯¯
Step 5: Plot the points from the table on the same coordinate plane.
Use the x-values as the x-coordinates and the y-values as the y-coordinates.
Connect the plotted points to form the graph of the inverse equation.
Now, you have the graph of y = 2x + 3 and its inverse, y = (x - 3) / 2.
To graph the equation y = 2x + 3, we can use the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
In this case, the equation is already in the slope-intercept form, so you can directly identify the slope and y-intercept. The slope is 2, which means that for every unit increase in x, y increases by 2. The y-intercept is 3, which is the point (0, 3) on the graph.
To graph the equation, start by plotting the y-intercept (0, 3). From this point, use the slope to find additional points on the line. Since the slope is 2, you can move up 2 units and over 1 unit, or down 2 units and over 1 unit, to find additional points. For example, if you move up 2 units and over 1 unit from the y-intercept, you would get the point (1, 5). Similarly, you can find other points.
Once you have a few points, you can draw a straight line through them to represent the equation y = 2x + 3.
Now, let's find the inverse of the equation.
To find the inverse, we need to swap the x and y variables and solve for y. The inverse of y = 2x + 3 can be written as x = 2y + 3.
Next, solve the inverse equation for y to rewrite it in the slope-intercept form. Subtract 3 from both sides:
x - 3 = 2y
Then, divide both sides by 2:
(x - 3) / 2 = y
The inverse equation, in slope-intercept form, is y = (x - 3) / 2.
To graph the inverse, follow the same steps as before. Start by identifying the y-intercept, which is the point (0, -1.5), and then use the slope to find more points.
Draw a straight line passing through the points to represent the inverse equation y = (x - 3) / 2.
Note that we can observe that the original equation and its inverse are reflections of each other across the line y = x.