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Chapter 1: Problem 27

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=|3 x-7|\)

Short Answer

Expert verified
The x-intercept is at \(x = \frac{7}{3}\) and the y-intercept is at y = 7.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation, and solve for x. Therefore, the equation becomes: 0 = |3x-7|.
02

Calculate the x-intercept

When the absolute value of a number is 0, that number itself must be 0. Therefore, the equation becomes: 3x - 7 = 0. Solving for x, this provides \(x = \frac{7}{3}\). This gives the x-intercept of the function.
03

Find the y-intercept

To find the y-intercept, set x to 0 in the equation, and solve for y. The equation becomes: y = |3*0 - 7|. Solving this equation results in y = |-7|= 7.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Absolute Value Function
The absolute value function is a fascinating concept in mathematics. It is often represented as \(|x|\), which means the distance of a number \(x\) from zero on the number line.
  • The absolute value is always non-negative.
  • For a positive number or zero, the absolute value returns the number itself.
  • For a negative number, it returns the positive counterpart.
In the given equation, \(y = |3x - 7|\), the absolute value influences how the graph behaves. When solving problems with absolute values, it is crucial to remember that the expression inside the brackets could be either positive or negative. This influences whether the equation needs to be split into separate cases during solving.
Solving Equations Involving Absolute Values
Solving equations with absolute values often requires a few extra steps compared to regular linear equations. When encountering an absolute value equation, such as \(|3x - 7| = 0\), we think of it as solving two scenarios:
  • The positive scenario, \(3x - 7 = 0\): This equation follows directly from the absolute value definition. Solving it helps find the x-intercepts such as in this problem's solution.
  • The zero value scenario, because the problem's step involves setting the absolute value equal to zero. Here, \(3x - 7\) should be zero itself.
These steps ensure no solutions are overlooked. The resulting x-intercept in this exercise is \(x = \frac{7}{3}\), where the graph touches or crosses the x-axis.
Application of Coordinate Geometry
Coordinate geometry, or Cartesian geometry, allows us to visualize and analyze relationships and patterns. The x- and y-intercepts are particularly important in graphing. For the equation \(y = |3x - 7|\), the intercepts tell us specific points where the graph crosses the axes.
  • x-intercept: Solved by setting \(y \) to zero, which yielded \(x = \frac{7}{3}\). This point \(\left(\frac{7}{3}, 0\right)\) is where the graph intersects the x-axis.
  • y-intercept: Found by setting \(x \) to zero, resulting in \(y = 7\). The graph crosses the y-axis at the point \((0, 7)\).
These intercepts help in sketching the graph, marking vital transition points, and understanding how the function behaves across the Cartesian plane.

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Most popular questions from this chapter

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