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Chapter 3: Problem 79

In your own words, explain how to find \(x\) - and \(y\) -intercepts.

Short Answer

Expert verified
Set y=0 to find x-intercepts; set x=0 to find y-intercepts.

Step by step solution

01

Understanding Intercepts

The x-intercept of a graph is the point where the graph crosses the x-axis. This corresponds to a point where the y-coordinate is zero. Similarly, the y-intercept is the point where the graph crosses the y-axis, which occurs when the x-coordinate is zero.
02

Finding the X-Intercept

To find the x-intercept of a function, set the y-value to zero in the equation and solve for x. For example, if you have an equation of the line in the form of \( y = mx + b \), set \( y = 0 \) to find \( x \). The calculation will be: \( 0 = mx + b \), solve for \( x \): \( x = -\frac{b}{m} \).
03

Finding the Y-Intercept

To find the y-intercept, set the x-value to zero in the equation and solve for y. Using the same equation \( y = mx + b \), substitute \( x = 0 \): \( y = m imes 0 + b \), which simplifies to \( y = b \).
04

Conclusion of Intercepts

After calculating both x and y-intercepts, you have identified the points where the graph crosses the x-axis and y-axis. These are fundamental characteristics of linear equations useful in sketching and understanding the behavior of the graph.

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Key Concepts

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

Understanding the X-Intercept
The x-intercept is a crucial point in the graph of a function where it crosses the x-axis. At this point, the y-value is always zero. This means to find the x-intercept, you need to determine where the function equals zero on the y-axis. An easy way to do this in linear equations, such as in the form of \( y = mx + b \), is by setting \( y = 0 \) and solving for \( x \).

This involves the following simple steps:
  • Set \( y = 0 \) in the equation \( y = mx + b \).
  • Solve the resulting equation \( 0 = mx + b \) for \( x \).
  • Rearrange to find \( x = -\frac{b}{m} \).

These steps will guide you to the x-coordinate of the point where the graph intersects the x-axis. It's a straightforward method to apply, making it a typically simple process to find the x-intercept.
Understanding the Y-Intercept
The y-intercept of a graph is where the line meets the y-axis. At this intersection, the x-value equals zero. This is one of the constants you can find directly from the equation of a line.

For linear equations like \( y = mx + b \), finding the y-intercept is quite simple:

  • Substitute \( x = 0 \) into the equation.
  • After substitution, the equation simplifies as \( y = m \cdot 0 + b \).
  • This results in \( y = b \), meaning \( b \) is the y-intercept.

Understanding the y-intercept forms part of sketching a graph, as it tells you where the line crosses the y-axis. In practical terms, this value could represent an initial quantity when all other factors are zero, like a starting value in some contexts.
Introduction to Linear Equations
Linear equations are the simplest form of equations that have a straight-line graph. They are often written in the form \( y = mx + b \), known as the slope-intercept form. Here, \( m \) is the slope of the line, and \( b \) is the y-intercept. These equations are essential in mathematics as they help model simple relationships between two variables.

Some core aspects of linear equations include:
  • **Slope (\( m \)):** Indicates the steepness of the line. A positive slope means the line ascends from left to right, whereas a negative slope descends.
  • **Y-Intercept (\( b \)):** Shows where the line crosses the y-axis. This is critical for both graphing and interpreting linear relationships.
  • **Intersections:** Understanding how the line intersects axes is key in graph analysis; this introduces the concepts of x and y intercepts in practical applications.

The simplicity of linear equations makes them powerful tools in various fields such as economics, physics, and everyday problem-solving scenarios.

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Most popular questions from this chapter

The function \(f(x)=0.42 x+10.5,\) can be used to predict diamond production. For this function, \(x\) is the number of years after \(2000,\) and \(f(x)\) is the value (in billions of dollars) of the years diamond production. Use the function to predict diamond production in 2012 .

From the Chapter 3 opener, we have two functions to describe the percent of college students taking at least one online course. For both functions, \(x\) is the number of years since 2000 and \(y\) (or \(f(x)\) or \(g(x))\) is the percent of students taking at least one online course. $$f(x)=2.7 x+4.1 \text { or } g(x)=0.07 x^{2}+1.9 x+5.9$$ Use Exercises \(81-84\) and compare \(f(9)\) and \(g(9),\) then \(f(16)\) and \(g(16) .\) As \(x\) increases, are the function values staying about the same or not? Explain your answer.

Find an equation of the perpendicular bisector of the line segment whose endpoints are given. \((-6,8) ;(-4,-2)\)

Broyhill Furniture found that it takes 2 hours to manufacture each table for one of its special dining room sets. Each chair takes 3 hours to manufacture. A total of 1500 hours is available to produce tables and chairs of this style. The linear equation that models this situation is \(2 x+3 y=1500\) where \(x\) represents the number of tables produced and \(y\) the number of chairs produced. a. Complete the ordered pair solution \((0,)\) of this equation. Describe the manufacturing situation this solution corresponds to. b. Complete the ordered pair solution \((, 0)\) for this equation. Describe the manufacturing situation this solution corresponds to. c. If 50 tables are produced, find the greatest number of chairs the company can make.

Use the graph of the functions below to answer Exercises 69 through \(80 . (GRAPH CANNOT COPY) If \)f(1)=-10,$ write the corresponding ordered pair.
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