Which Linear Functions Represent A Slope Of 4? Check All That Apply.Table 1:${ \begin{tabular}{|c|c|} \hline X X X & Y Y Y \ \hline 3 & -11 \ \hline 6 & 1 \ \hline 9 & 13 \ \hline 12 & 25 \ \hline \end{tabular} }$Table
Which Linear Functions Represent a Slope of 4? Check All That Apply
In mathematics, a linear function is a polynomial function of degree one or less. It is a function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. The slope of a linear function represents the rate of change of the function with respect to the input variable. In this article, we will explore which linear functions represent a slope of 4.
Understanding Slope
The slope of a linear function is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope can be positive, negative, or zero, depending on the direction and steepness of the line.
Calculating Slope
To calculate the slope of a linear function, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Analyzing Table 1
Let's analyze the data in Table 1 to determine which linear functions represent a slope of 4.
| x | y |
|---|---|
| 3 | -11 |
| 6 | 1 |
| 9 | 13 |
| 12 | 25 |
We can calculate the slope using the formula above. Let's choose two points, (3, -11) and (6, 1).
m = (1 - (-11)) / (6 - 3) m = 12 / 3 m = 4
This means that the linear function passing through the points (3, -11) and (6, 1) has a slope of 4.
Checking Other Points
Let's check the other points in Table 1 to see if they also represent a slope of 4.
| x | y |
|---|---|
| 6 | 1 |
| 9 | 13 |
We can calculate the slope using the formula above. Let's choose two points, (6, 1) and (9, 13).
m = (13 - 1) / (9 - 6) m = 12 / 3 m = 4
This means that the linear function passing through the points (6, 1) and (9, 13) also has a slope of 4.
| x | y |
|---|---|
| 9 | 13 |
| 12 | 25 |
We can calculate the slope using the formula above. Let's choose two points, (9, 13) and (12, 25).
m = (25 - 13) / (12 - 9) m = 12 / 3 m = 4
This means that the linear function passing through the points (9, 13) and (12, 25) also has a slope of 4.
| x | y |
|---|---|
| 12 | 25 |
| 3 | -11 |
We can calculate the slope using the formula above. Let's choose two points, (12, 25) and (3, -11).
m = (-11 - 25) / (3 - 12) m = -36 / -9 m = 4
This means that the linear function passing through the points (12, 25) and (3, -11) also has a slope of 4.
In conclusion, the linear functions passing through the points (3, -11) and (6, 1), (6, 1) and (9, 13), (9, 13) and (12, 25), and (12, 25) and (3, -11) all represent a slope of 4. These linear functions can be written in the form of y = 4x + b, where b is the y-intercept.
The final answer is:
- y = 4x - 47
- y = 4x - 35
- y = 4x - 23
- y = 4x - 11
Note: The y-intercept (b) is different for each linear function, but the slope (m) is the same, which is 4.
Q&A: Linear Functions with a Slope of 4
In our previous article, we explored which linear functions represent a slope of 4. We analyzed the data in Table 1 and determined that the linear functions passing through the points (3, -11) and (6, 1), (6, 1) and (9, 13), (9, 13) and (12, 25), and (12, 25) and (3, -11) all have a slope of 4. In this article, we will answer some frequently asked questions about linear functions with a slope of 4.
Q: What is the equation of a linear function with a slope of 4?
A: The equation of a linear function with a slope of 4 is y = 4x + b, where b is the y-intercept.
Q: How do I find the y-intercept (b) of a linear function with a slope of 4?
A: To find the y-intercept (b) of a linear function with a slope of 4, you can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. For example, if we know that the point (3, -11) is on the line, we can plug in the values to get:
y - (-11) = 4(x - 3) y + 11 = 4x - 12 y = 4x - 23
So, the y-intercept (b) is -23.
Q: Can I use the slope-intercept form (y = mx + b) to find the y-intercept (b) of a linear function with a slope of 4?
A: Yes, you can use the slope-intercept form (y = mx + b) to find the y-intercept (b) of a linear function with a slope of 4. For example, if we know that the slope (m) is 4, we can plug in the values to get:
y = 4x + b
To find the y-intercept (b), we can use the point (3, -11) and plug in the values:
-11 = 4(3) + b -11 = 12 + b b = -23
So, the y-intercept (b) is -23.
Q: How do I graph a linear function with a slope of 4?
A: To graph a linear function with a slope of 4, you can use the slope-intercept form (y = mx + b) and plot two points on the line. For example, if we know that the slope (m) is 4 and the y-intercept (b) is -23, we can plot the points (0, -23) and (3, -11) on the coordinate plane.
Q: Can I use a calculator to graph a linear function with a slope of 4?
A: Yes, you can use a calculator to graph a linear function with a slope of 4. Most graphing calculators have a built-in function to graph linear equations in the form of y = mx + b. Simply enter the values of m and b, and the calculator will graph the line for you.
In conclusion, linear functions with a slope of 4 can be written in the form of y = 4x + b, where b is the y-intercept. We can find the y-intercept (b) by using the point-slope form of a linear equation or the slope-intercept form (y = mx + b). We can also graph a linear function with a slope of 4 by plotting two points on the line or using a calculator.
The final answer is:
- y = 4x - 47
- y = 4x - 35
- y = 4x - 23
- y = 4x - 11
Note: The y-intercept (b) is different for each linear function, but the slope (m) is the same, which is 4.