Quadrants and the Graph of the Linear Function

Oct 13, 2025 by recoveryshake.com 104 views

In mathematics, a linear function is a polynomial function of degree one or less. It is a simple function that can be represented in the form of y = mx + b, where m is the slope and b is the y-intercept. The graph of a linear function is a straight line that can be either positive, negative, or zero. In this article, we will consider the graph of the linear function h(x) = -6 + \frac{2}{3}x and determine which quadrant it will not go through.

The Slope and Y-Intercept of the Linear Function

The given linear function is h(x) = -6 + \frac{2}{3}x. To understand the graph of this function, we need to analyze its slope and y-intercept. The slope of the function is \frac{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.

The y-intercept of the function is -6, which is a negative value. This means that the graph of the function will intersect the y-axis at a point below the origin. In other words, the graph will start below the x-axis and rise upwards.

Quadrants and the Graph of the Linear Function

A quadrant is a region of the coordinate plane that is divided by the x-axis and the y-axis. There are four quadrants in total: Quadrant I, Quadrant II, Quadrant III, and Quadrant IV.

Quadrant I is the region where both x and y are positive. Quadrant II is the region where x is negative and y is positive. Quadrant III is the region where both x and y are negative. Quadrant IV is the region where x is positive and y is negative.

Which Quadrant Will the Graph Not Go Through?

To determine which quadrant the graph of the linear function will not go through, we need to analyze the slope and y-intercept of the function. As mentioned earlier, the slope of the function is positive, indicating that the graph will rise from left to right. This means that the graph will not go through Quadrant II, where x is negative and y is positive.

The graph will start below the x-axis and rise upwards, intersecting the x-axis at a point to the right of the origin. This means that the graph will not go through Quadrant II, because the slope is positive and the y-intercept is negative.

Conclusion

In conclusion, the graph of the linear function h(x) = -6 + \frac{2}{3}x will not go through Quadrant II. This is because the slope of the function is positive, indicating that the graph will rise from left to right, and the y-intercept is negative, indicating that the graph will start below the x-axis.

Why the Graph Will Not Go Through Quadrant II

The graph will not go through Quadrant II because the slope is positive and the y-intercept is negative. This means that the graph will rise from left to right, intersecting the x-axis at a point to the right of the origin. As a result, the graph will not go through Quadrant II, where x is negative and y is positive.

Key Takeaways

The graph of a linear function is a straight line that can be either positive, negative, or zero.
The slope of a linear function determines the direction of the graph.
The y-intercept of a linear function determines the starting point of the graph.
Quadrant II is the region where x is negative and y is positive.
The graph of the linear function h(x) = -6 + \frac{2}{3}x will not go through Quadrant II.

Final Thoughts

In the previous article, we discussed the graph of the linear function h(x) = -6 + \frac{2}{3}x and determined which quadrant it will not go through. In this article, we will answer some frequently asked questions (FAQs) about the graph of a linear function.

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

A: A linear function is a polynomial function of degree one or less, 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: How do I determine the slope of a linear function?

A: To determine the slope of a linear function, you need to look at the coefficient of the x-term. If the coefficient is positive, the slope is positive. If the coefficient is negative, the slope is negative.

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

A: The y-intercept of a linear function is the point where the graph intersects the y-axis. It is the value of y when x is equal to zero.

Q: How do I determine which quadrant a linear function will go through?

A: To determine which quadrant a linear function will go through, you need to analyze the slope and y-intercept of the function. If the slope is positive and the y-intercept is positive, the graph will go through Quadrant I. If the slope is negative and the y-intercept is positive, the graph will go through Quadrant IV. If the slope is positive and the y-intercept is negative, the graph will go through Quadrant III. If the slope is negative and the y-intercept is negative, the graph will go through Quadrant II.

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

A: Yes, a linear function can have a negative slope and a positive y-intercept. In this case, the graph will go through Quadrant IV.

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

A: Yes, a linear function can have a positive slope and a negative y-intercept. In this case, the graph will go through Quadrant III.

Q: How do I graph a linear function?

A: To graph a linear function, you need to plot two points on the graph. One point is the y-intercept, and the other point is a point on the graph that is not the y-intercept. Then, you need to draw a straight line through the two points.

Q: Can a linear function have a zero slope?

A: Yes, a linear function can have a zero slope. In this case, the graph will be a horizontal line.

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

A: Yes, a linear function can have a zero y-intercept. In this case, the graph will be a vertical line.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about the graph of a linear function. We have discussed the difference between a linear function and a quadratic function, how to determine the slope of a linear function, and how to determine which quadrant a linear function will go through. We have also discussed how to graph a linear function and whether a linear function can have a zero slope or a zero y-intercept.

Key Takeaways

A linear function is a polynomial function of degree one or less.
The graph of a linear function is a straight line.
The slope of a linear function determines the direction of the graph.
The y-intercept of a linear function determines the starting point of the graph.
A linear function can have a negative slope and a positive y-intercept.
A linear function can have a positive slope and a negative y-intercept.
A linear function can have a zero slope or a zero y-intercept.

Final Thoughts

In this article, we have discussed the graph of a linear function and answered some frequently asked questions (FAQs) about it. We have provided information on how to determine the slope and y-intercept of a linear function, how to graph a linear function, and whether a linear function can have a zero slope or a zero y-intercept. We hope that this information has been helpful to you.