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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 of the equation \(y = |3x-7|\) is at \((7/3, 0)\). The y- intercept is at \((0, 7)\).

Step by step solution

01

- Finding the x-intercept

To find the x-intercept, set \(y = 0\) in the given equation \[y=|3x-7|\]. This gives the equation \(0 = |3x-7|\). According to the properties of the absolute value function, this is true when \((3x - 7) = 0\). Solving this equation gives us the value \(x = 7/3\).
02

- Finding the y-intercept

To find the y-intercept, set \(x = 0\) in the given modulus equation: \[y = |3*0 - 7|\]. This implies \(y = |-7|\), but since the absolute value of a negative number is its positive counterpart, \(y = 7\). The y-intercept is at (0,7).

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Key Concepts

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

Understanding x-intercepts in Absolute Value Functions
The x-intercept of a function is the point where the graph intersects the x-axis. This means that at the x-intercept, the y-value will be zero.
To find the x-intercept for the equation \( y = |3x - 7| \), we start by setting \( y \) to zero. This gives us the equation \( 0 = |3x - 7| \).
  • An important property of absolute value functions is that the expression inside the absolute value will equal zero to satisfy the equation.
  • For our case, solving \( 3x - 7 = 0 \) gives us \( x = \frac{7}{3} \).
This result, \( x = \frac{7}{3} \), is the x-coordinate at which the function's graph crosses the x-axis. Hence, the x-intercept for the graph of this equation is at the point \( (\frac{7}{3}, 0) \).
Understanding where the function crosses the x-axis is crucial in sketching and analyzing the graph.
Understanding y-intercepts in Absolute Value Functions
The y-intercept of a function is where the graph crosses the y-axis. At this point, the x-value is zero.
To find the y-intercept of \( y = |3x - 7| \), we substitute \( x=0 \) into the equation: \( y = |3 * 0 - 7| \).
  • This simplifies to \( y = |-7| \), and since the absolute value function returns the positive version of a number, \( y = 7 \).
  • This means the y-intercept is located at the point \( (0,7) \).
Knowing the y-intercept helps us understand the graph's upward or downward shift along the y-axis, providing a reference point when drawing or estimating the graph.
Y-intercepts are important in determining the starting point of a graph when sketching it by hand.
Graphing Absolute Value Equations
Graphing absolute value equations, like \( y = |3x - 7| \), involves understanding the unique V-shape characteristic of absolute value functions.
  • The absolute value function reflects all negative outputs into positive outputs, creating a distinctive shape.
  • The vertex of this absolute value function is at the point \( (\frac{7}{3}, 0) \), determined when \( 3x - 7 = 0 \).
  • This vertex tells us where the V-shape changes direction.
From the vertex, the graph rises steeply to the right and left along the lines \( y = 3x - 7 \) and \( y = -(3x - 7) \), respectively.
Knowing the x- and y-intercepts helps in plotting these points accurately.
Additionally, by locating the intercepts and vertex, you can sketch a rough graph, assisting in visualizing how changes in the equation's parameters shift the graph horizontally or vertically.
Ensure each significant point, like the x-intercept, y-intercept, and vertex, is correctly marked for an accurate graph representation.

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