Find The X-Intercept And Y-Intercept Of Y=|3x-6|

In the realm of mathematics, absolute value functions play a crucial role in understanding the magnitude of numbers and their relationships. The absolute value of a number is its distance from zero, regardless of its sign. This concept forms the foundation for a unique class of functions that exhibit distinct graphical properties, such as V-shaped curves and sharp corners.

Understanding Absolute Value Functions: In this article, we delve into the intricacies of an absolute value function, specifically $y = |3x - 6|$. Our primary objective is to pinpoint the precise coordinates of the x-intercept and y-intercept, which serve as pivotal points in characterizing the function's behavior. To achieve this, we'll systematically explore the function's properties, employing algebraic techniques and graphical interpretations.

The Significance of Intercepts: Intercepts, in the context of functions, hold paramount importance. The x-intercept, also known as the root or zero of the function, represents the point where the graph intersects the x-axis. At this point, the y-coordinate is invariably zero. Conversely, the y-intercept signifies the point where the graph intersects the y-axis, with the x-coordinate being zero. These intercepts provide valuable insights into the function's nature, revealing its behavior near the axes and its overall trajectory.

Embarking on the Quest for Coordinates: Our journey to determine the intercepts of $y = |3x - 6|$ begins with a strategic approach. We'll first focus on finding the x-intercept, where the function's value equals zero. This involves setting the expression inside the absolute value bars to zero and solving for x. Subsequently, we'll tackle the y-intercept, where the function's graph intersects the y-axis. This entails substituting x = 0 into the function and evaluating the resulting y-value.

Finding the X-Intercept: Where the Function Meets the Axis

To embark on our quest for the x-intercept, we must first grasp the fundamental concept that the x-intercept is the point where the function's graph gracefully intersects the x-axis. At this pivotal juncture, the y-coordinate takes on the value of zero. Therefore, to pinpoint the x-intercept of our absolute value function, $y = |3x - 6|$, we embark on a mathematical expedition where we set the function's value to zero, thereby transforming the equation into a solvable puzzle.

The equation we seek to conquer is $|3x - 6| = 0$. This equation, at first glance, may appear enigmatic, but beneath its seemingly complex façade lies a straightforward solution. The absolute value of an expression, by its very nature, is its distance from zero, irrespective of its sign. Consequently, the only way for the absolute value of an expression to equal zero is if the expression itself is zero.

Thus, we arrive at the crucial realization that $3x - 6$ must be equal to zero. This transforms our absolute value equation into a linear equation, a familiar terrain in the world of mathematics. To solve for x, we employ the fundamental principles of algebraic manipulation. We add 6 to both sides of the equation, effectively isolating the term containing x. This yields the equation $3x = 6$.

Our journey continues as we seek to isolate x completely. To achieve this, we divide both sides of the equation by 3, the coefficient that currently clings to x. This division unveils the value of x, revealing that $x = 2$. This triumphant result signifies that the x-intercept of the function $y = |3x - 6|$ lies at the point where x equals 2.

To express this x-intercept in its complete glory, we present it as an ordered pair, a harmonious duo of coordinates that pinpoint its location on the Cartesian plane. The x-intercept, therefore, is represented as (2, 0). This ordered pair serves as a beacon, guiding us to the precise point where the function's graph gracefully kisses the x-axis.

Unveiling the Y-Intercept: The Function's Dance with the Vertical Axis

Having successfully located the x-intercept, our mathematical expedition now guides us towards the y-intercept, another crucial landmark in the function's landscape. The y-intercept, like its x-counterpart, holds a special significance, marking the point where the function's graph elegantly intersects the y-axis. At this point of intersection, the x-coordinate gracefully bows out, assuming the value of zero.

To unearth the y-intercept of our absolute value function, $y = |3x - 6|$, we embark on a similar quest, but this time, we set the x-coordinate to zero. This substitution transforms the function's equation into a simpler form, allowing us to isolate the y-coordinate and unveil the y-intercept's secret location.

Substituting $x = 0$ into the equation, we find ourselves face-to-face with $y = |3(0) - 6|$. This substitution effectively eliminates the x term, leaving us with an expression that solely depends on the constant term within the absolute value bars. Simplifying the expression, we arrive at $y = |-6|$.

The absolute value of a number, as we've previously established, is its distance from zero, irrespective of its sign. Therefore, the absolute value of -6 is simply 6. This revelation unveils the y-coordinate of the y-intercept, revealing that $y = 6$. This triumphant result signifies that the y-intercept of the function $y = |3x - 6|$ lies at the point where y equals 6.

To express this y-intercept in its full glory, we once again present it as an ordered pair, a harmonious duo of coordinates that pinpoint its location on the Cartesian plane. The y-intercept, therefore, is represented as (0, 6). This ordered pair serves as another beacon, guiding us to the precise point where the function's graph gracefully dances with the y-axis.

Completing the Sentence: A Synthesis of Our Findings

Having diligently navigated the mathematical terrain and successfully unearthed both the x-intercept and the y-intercept, we now stand poised to complete the sentence that initiated our expedition. The sentence, a testament to our quest, seeks to capture the essence of these intercepts, solidifying our understanding of the function's behavior.

The x-intercept of the function, as we've definitively established, lies at the point (2, 0). This ordered pair serves as a tangible representation of the function's dance with the x-axis, marking the spot where the graph gracefully crosses the horizontal plane. Similarly, the y-intercept, our second triumphant discovery, resides at the point (0, 6). This ordered pair embodies the function's embrace of the y-axis, pinpointing the location where the graph gracefully intersects the vertical plane.

Therefore, to complete the sentence with precision and accuracy, we fill in the blanks with the coordinates we've so meticulously uncovered. The completed sentence, a testament to our mathematical prowess, reads as follows: The x-intercept of the function is located at (2, 0) and the y-intercept of the function is located at (0, 6).

Graphing the Function: A Visual Confirmation of Our Results

To further solidify our understanding of the function $y = |3x - 6|$ and visually corroborate our findings regarding the intercepts, we embark on a graphical representation of the function. Graphing the function provides a powerful visual aid, allowing us to observe the function's behavior, its intercepts, and its overall trajectory on the Cartesian plane.

The graph of an absolute value function typically exhibits a distinctive V-shape, characterized by a sharp corner or vertex. This V-shape arises from the absolute value's property of reflecting negative values to their positive counterparts, creating a symmetrical pattern around the vertex.

To plot the graph of $y = |3x - 6|$, we can begin by identifying the vertex, which represents the point where the function changes direction. The vertex of an absolute value function in the form $y = |ax + b|$ occurs at the point where $ax + b = 0$. In our case, this translates to $3x - 6 = 0$, which we've already solved to find $x = 2$. Substituting this value back into the function, we find $y = |3(2) - 6| = 0$. Therefore, the vertex of our function resides at the point (2, 0), which, notably, coincides with the x-intercept we previously determined.

Having located the vertex, we can now plot additional points to trace the function's V-shape. We can choose values of x both to the left and right of the vertex, calculate the corresponding y-values, and plot these points on the graph. For instance, if we choose $x = 0$, we find $y = |3(0) - 6| = 6$, which corresponds to the y-intercept we previously identified. Similarly, if we choose $x = 4$, we find $y = |3(4) - 6| = 6$.

By plotting these points and connecting them with smooth lines, we can visualize the graph of $y = |3x - 6|$. The graph clearly exhibits the V-shape characteristic of absolute value functions, with the vertex at (2, 0) and the y-intercept at (0, 6). This visual representation serves as a compelling confirmation of our algebraic calculations, reinforcing our understanding of the function's intercepts and its overall behavior.

In this comprehensive exploration, we've embarked on a mathematical journey to unravel the mysteries of the absolute value function $y = |3x - 6|$. Our quest centered around pinpointing the precise coordinates of the x-intercept and y-intercept, pivotal points that illuminate the function's behavior and its relationship with the coordinate axes.

Through a meticulous application of algebraic techniques and graphical interpretations, we successfully determined that the x-intercept of the function resides at the point (2, 0), while the y-intercept graces the coordinate plane at the point (0, 6). These coordinates, expressed as ordered pairs, serve as tangible representations of the function's interactions with the x and y axes, marking the points where the graph gracefully intersects these fundamental lines.

Our journey began with a fundamental understanding of absolute value functions and the significance of intercepts. We then delved into the specific equation at hand, employing algebraic manipulation to solve for the x-intercept by setting the function's value to zero. Subsequently, we tackled the y-intercept, substituting x = 0 into the equation and evaluating the resulting y-value.

To further solidify our understanding and validate our findings, we ventured into the realm of graphical representation. By plotting the function on the Cartesian plane, we visually confirmed the location of the intercepts and observed the characteristic V-shape of absolute value functions. The graph served as a powerful testament to the accuracy of our algebraic calculations, reinforcing our grasp of the function's behavior.

Our exploration culminated in the triumphant completion of the sentence that sparked our mathematical endeavor. By filling in the blanks with the coordinates we meticulously uncovered, we articulated the precise location of the intercepts, solidifying our understanding of the function's properties. The completed sentence stands as a testament to our mathematical prowess, capturing the essence of our exploration.

In essence, this journey through the absolute value function $y = |3x - 6|$ has not only provided us with the specific coordinates of the intercepts but has also deepened our appreciation for the elegance and power of mathematical tools. The ability to analyze functions, determine their key features, and represent them graphically is a cornerstone of mathematical understanding, and this exploration has served as a valuable exercise in honing these skills. The x-intercept of the function is located at (2, 0) and the y-intercept of the function is located at (0, 6).