Graph Of Linear Function H(x) = -6 + (2/3)x Which Quadrant Not Go Through

In the realm of mathematics, linear functions play a crucial role. They are characterized by their straight-line graphs and constant rate of change. Understanding the properties of linear functions, such as their slope and intercepts, is fundamental to predicting their behavior and the regions of the coordinate plane they traverse. Let's delve into the specifics of linear functions and how they relate to quadrants.

A linear function is typically expressed in the slope-intercept form: y = mx + b, where 'm' represents the slope, and 'b' is the y-intercept. The slope determines the steepness and direction of the line. A positive slope indicates an increasing line, while a negative slope signifies a decreasing line. The y-intercept is the point where the line intersects the y-axis.

The coordinate plane is divided into four regions known as quadrants, numbered I through IV, starting from the upper right quadrant and proceeding counter-clockwise. Each quadrant is defined by the signs of the x and y coordinates:

• Quadrant I: x > 0, y > 0 (Positive x, Positive y)
• Quadrant II: x < 0, y > 0 (Negative x, Positive y)
• Quadrant III: x < 0, y < 0 (Negative x, Negative y)
• Quadrant IV: x > 0, y < 0 (Positive x, Negative y)

By analyzing the slope and y-intercept of a linear function, we can predict which quadrants its graph will pass through. This understanding is crucial for visualizing the function's behavior and its relationship to the coordinate plane.

Let's turn our attention to the specific linear function given: h(x) = -6 + (2/3)x. Our primary goal is to determine which quadrant the graph of this function will not pass through and to provide a clear rationale for our conclusion. To achieve this, we need to carefully analyze the function's slope and y-intercept.

First, let's rewrite the function in the standard slope-intercept form, y = mx + b, where 'm' is the slope and 'b' is the y-intercept. By rearranging the terms, we get:

y = (2/3)x - 6

Now, we can easily identify the slope and y-intercept:

• Slope (m): 2/3
• Y-intercept (b): -6

The slope, 2/3, is a positive value. This tells us that the line is increasing, meaning that as x increases, y also increases. The y-intercept, -6, is a negative value. This indicates that the line intersects the y-axis at the point (0, -6).

With this information, we can begin to visualize the graph of the function. The line starts at the point (0, -6) and rises as it moves from left to right. This means that the line will definitely pass through Quadrant IV (where x is positive and y is negative) and Quadrant I (where both x and y are positive).

To determine whether the line passes through the other quadrants, we need to consider its behavior as x takes on negative values. As x becomes increasingly negative, y will also decrease, but the rate of decrease is determined by the slope. Since the slope is positive, the line will eventually cross the x-axis and enter Quadrant I. However, it's important to examine whether the line will ever enter Quadrant II.

Now, let's pinpoint the quadrant that the graph of h(x) = -6 + (2/3)x will not pass through. We've already established that the line traverses Quadrants I and IV. To determine if it passes through Quadrants II and III, let's consider the characteristics of each quadrant and how they relate to our function's properties.

• Quadrant II: In Quadrant II, x is negative, and y is positive. For the line to pass through Quadrant II, there must be a point where a negative x-value produces a positive y-value. Let's examine this possibility.

  We have the equation y = (2/3)x - 6. To find the x-value that corresponds to y = 0 (the x-intercept), we set y to 0 and solve for x:

  0 = (2/3)x - 6

  6 = (2/3)x

  x = 9

  This tells us that the line crosses the x-axis at x = 9, which is in Quadrant IV. Now, let's consider negative x-values. As x becomes more negative, the term (2/3)x will also become more negative. Since we are subtracting 6 from this term, y will always be negative for negative x-values. Therefore, the line will not enter Quadrant II.

• Quadrant III: In Quadrant III, both x and y are negative. As we established earlier, for negative x-values, y will also be negative. Therefore, the line will pass through Quadrant III.

Based on our analysis, we can confidently conclude that the graph of h(x) = -6 + (2/3)x will not pass through Quadrant II. This is because the positive slope and negative y-intercept cause the line to increase from a negative y-value to positive values as x increases, effectively bypassing the region where x is negative and y is positive.

In summary, the graph of the linear function h(x) = -6 + (2/3)x will not go through Quadrant II. The reasoning behind this lies in the function's slope and y-intercept. The positive slope indicates that the line increases as x increases, while the negative y-intercept means that the line starts below the x-axis. This combination ensures that the line will pass through Quadrants I, III, and IV, but it will never enter Quadrant II, where x is negative and y is positive. Understanding how the slope and y-intercept influence a linear function's graph is a fundamental concept in mathematics, allowing us to predict its behavior and its relationship with the coordinate plane quadrants.

By carefully analyzing the equation and visualizing its graph, we can confidently determine the quadrants it will traverse, gaining a deeper appreciation for the interplay between algebraic expressions and geometric representations.