The Graph of the Linear Function

Oct 13, 2025 by magicalceylontours.com 110 views

In mathematics, the concept of quadrants is crucial when dealing with linear functions. A linear function is a polynomial function of degree one, and it can be represented in the form of y = mx + b, where m is the slope and b is the y-intercept. In this article, we will explore the graph of the linear function h(x) = -6 + 2/3x and determine which quadrant it will not go through.

What are Quadrants?

Quadrants are the four sections of the Cartesian coordinate system. They are labeled as I, II, III, and IV, and they are divided by the x-axis and the y-axis. The x-axis and y-axis intersect at the origin (0, 0), which is the point where both the x and y coordinates are zero.

Quadrant I: This quadrant includes all the points where both the x and y coordinates are positive.
Quadrant II: This quadrant includes all the points where the x coordinate is negative and the y coordinate is positive.
Quadrant III: This quadrant includes all the points where both the x and y coordinates are negative.
Quadrant IV: This quadrant includes all the points where the x coordinate is positive and the y coordinate is negative.

The Graph of the Linear Function

The given linear function is h(x) = -6 + 2/3x. To determine which quadrant the graph will not go through, we need to analyze the slope and the y-intercept of the function.

Slope: The slope of the function is 2/3, which is a positive value. This means that the graph of the function will have a positive slope, indicating that it will rise from left to right.
Y-intercept: The y-intercept of the function is -6, which is a negative value. This means that the graph of the function will cross the y-axis at a point below the origin.

Which Quadrant Will the Graph Not Go Through?

Based on the analysis of the slope and the y-intercept, we can conclude that the graph of the function h(x) = -6 + 2/3x will not go through Quadrant I.

Why Quadrant I? The graph will not go through Quadrant I because the y-intercept is negative. The y-intercept is the point where the graph crosses the y-axis, and in Quadrant I, both the x and y coordinates are positive. Since the y-intercept is negative, the graph will not enter Quadrant I.

Conclusion

In conclusion, the graph of the linear function h(x) = -6 + 2/3x will not go through Quadrant I. This is because the y-intercept is negative, and the graph will cross the y-axis at a point below the origin. The graph will have a positive slope, indicating that it will rise from left to right, and it will enter Quadrants II, III, and IV.

Key Takeaways

The graph of a linear function can be represented in the form of y = mx + b, where m is the slope and b is the y-intercept.
Quadrants are the four sections of the Cartesian coordinate system, labeled as I, II, III, and IV.
The graph of a linear function will have a positive slope if the coefficient of x is positive.
The graph of a linear function will cross the y-axis at a point below the origin if the y-intercept is negative.

Frequently Asked Questions

Q: What is the slope of the linear function h(x) = -6 + 2/3x?

A: The slope of the linear function h(x) = -6 + 2/3x is 2/3.

Q: What is the y-intercept of the linear function h(x) = -6 + 2/3x?

A: The y-intercept of the linear function h(x) = -6 + 2/3x is -6.

Q: Which quadrant will the graph of the linear function h(x) = -6 + 2/3x not go through?

In this article, we will continue to explore the graph of the linear function h(x) = -6 + 2/3x and answer some frequently asked questions.

Q: What is the significance of the slope in a linear function?

A: The slope of a linear function is a measure of how steep the graph of the function is. It is calculated as the ratio of the change in y to the change in x. In the case of the linear function h(x) = -6 + 2/3x, the slope is 2/3, which means that for every unit increase in x, the value of y increases by 2/3 units.

Q: How do I determine the y-intercept of a linear function?

A: The y-intercept of a linear function is the point where the graph of the function crosses the y-axis. It is the value of y when x is equal to zero. In the case of the linear function h(x) = -6 + 2/3x, the y-intercept is -6, which means that the graph of the function crosses the y-axis at the point (0, -6).

Q: What is the relationship between the slope and the y-intercept of a linear function?

A: The slope and the y-intercept of a linear function are related in that they determine the direction and position of the graph of the function. The slope determines the steepness of the graph, while the y-intercept determines the position of the graph on the y-axis.

Q: Can a linear function have a negative slope?

A: Yes, a linear function can have a negative slope. This means that the graph of the function will have a negative steepness, indicating that it will fall from left to right.

Q: How do I graph a linear function?

A: To graph a linear function, you can use the slope-intercept form of the equation, which is y = mx + b, where m is the slope and b is the y-intercept. You can then plot the y-intercept on the y-axis and use the slope to determine the direction of the graph.

Q: What is the difference between a linear function and a quadratic function?

A: A linear function is a polynomial function of degree one, while a quadratic function is a polynomial function of degree two. The graph of a linear function is a straight line, while the graph of a quadratic function is a parabola.

Q: Can a linear function have a fractional slope?

A: Yes, a linear function can have a fractional slope. This means that the slope is a fraction, such as 2/3 or 3/4.

Q: How do I determine the equation of a linear function given its graph?

A: To determine the equation of a linear function given its graph, you can use the slope-intercept form of the equation, which is y = mx + b. You can then use the slope and the y-intercept to determine the equation of the function.

Q: What is the significance of the x-intercept of a linear function?

A: The x-intercept of a linear function is the point where the graph of the function crosses the x-axis. It is the value of x when y is equal to zero. In the case of the linear function h(x) = -6 + 2/3x, the x-intercept is 9, which means that the graph of the function crosses the x-axis at the point (9, 0).

Q: Can a linear function have a negative y-intercept?

A: Yes, a linear function can have a negative y-intercept. This means that the graph of the function will cross the y-axis at a point below the origin.

Q: How do I determine the equation of a linear function given its x-intercept and y-intercept?

A: To determine the equation of a linear function given its x-intercept and y-intercept, you can use the slope-intercept form of the equation, which is y = mx + b. You can then use the x-intercept and the y-intercept to determine the equation of the function.

Conclusion

Key Takeaways

The slope of a linear function determines the steepness of the graph.
The y-intercept of a linear function determines the position of the graph on the y-axis.
A linear function can have a negative slope or a negative y-intercept.
The x-intercept of a linear function is the point where the graph of the function crosses the x-axis.
A linear function can have a fractional slope.