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Chapter 2: Problem 114

For the following exercises, sketch a line with the given features. An \(x\) -intercept of \((-4,0)\) and \(y\) -intercept of \((0,-2)\)

Short Answer

Expert verified
Plot points at \((-4,0)\) and \((0,-2)\), then connect them with a line.

Step by step solution

01

Understand Intercepts

The given line has two intercepts: an \(x\)-intercept at \((-4,0)\) and a \(y\)-intercept at \((0,-2)\). The \(x\)-intercept is the point where the line crosses the \(x\)-axis, and the \(y\)-intercept is where it crosses the \(y\)-axis.
02

Plot the Intercepts

On a coordinate plane, plot the points \((-4, 0)\) and \((0, -2)\). These points indicate where the line crosses the respective axes.
03

Draw the Line

Connect the points \((-4, 0)\) and \((0, -2)\) with a straight line. This line represents the graph of the line passing through the given intercepts.

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Key Concepts

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

x-intercept
The term "x-intercept" refers to the point where a line crosses the x-axis on a graph. This means that at this specific point, the value of y is zero. To identify the x-intercept in an equation, substitute y with 0 and solve for x.

In practical terms, if you have an x-intercept of (-4, 0), it means that the line crosses the x-axis at the point where x equals -4. This point is crucial because it provides one of the key reference points needed to graph a line. Understanding intercepts helps us figure out how and where a line travels across the graph. If you can identify the x-intercept, graphing the line becomes much easier.
y-intercept
The y-intercept is similar to the x-intercept, but it marks where the line crosses the y-axis. At the y-intercept, the x-value is zero. To find this intercept from an equation, you set x to 0 and solve for y.

For a y-intercept located at (0, -2), this means the line crosses the y-axis when y equals -2. This point is another important reference that aids us in sketching a line.
  • The y-intercept provides insight into the starting or initial value of y when x is 0.
  • It tells us how the line behaves vertically on the coordinate plane.
  • Since it's a simple fix point, it's often used in calculations to quickly estimate where the line "begins" when plotted from the y-axis.
Both intercepts are fundamental in forming the equation of a line and are crucial for graph creation.
coordinate plane
The coordinate plane is like a map that helps us visualize equations and geometric figures. It consists of two perpendicular number lines, usually called the x-axis (horizontal) and y-axis (vertical).

Understanding how to navigate the coordinate plane is essential for graphing as it:
  • Allows us to plot points by providing a framework, known as coordinates, like (x, y).
  • Helps visualize relationships between variables and how they change.
  • Provides a space where lines and curves are easily represented spatially.
When working with intercepts, the coordinate plane assists in correctly placing these points, ensuring accurate line sketching when linking them up. Practicing with a coordinate plane builds your intuition for spatial relationships in mathematics.
graphing linear equations
Graphing linear equations involves visually laying out a line on the coordinate plane using known points such as intercepts. The process begins by finding points that the line will pass through, typically starting with x and y intercepts.

Steps to Graph Linear Equations:
  • Identify the intercepts of the equation. This can be done by solving the equation for when x (for the y-intercept) or y (for the x-intercept) is zero.
  • Plot these intercepts accurately on the coordinate plane.
  • Draw a straight line that passes through the points of intercepts. Ensure the line extends across the grid to include all possible values on the axes.
These steps transform a linear equation from abstract numbers and symbols into a tangible visual representation. This picture makes it easier to analyze and understand the behavior of the equation across different values. Becoming proficient in graphing linear equations is a foundational skill in algebra, paving the way for more advanced mathematical concepts.

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Most popular questions from this chapter

When hired at a new job selling electronics, you are given two pay options: \(\cdot\) Option A: Base salary of \(\$ 20,000\) a year with a commission of 12\(\%\) of your sales \(\cdot\) Option B: Base salary of \(\$ 26,000\) a year with a commission of 3\(\%\) of your sales How much electronics would you need to sell for option A to produce a larger income?

Determine whether the following function is increasing or decreasing. \(f(x)=-2 x+5\)

In \(1991,\) the moose population in a park was measured to be \(4,360 .\) By \(1999,\) the population was measured again to be \(5,880\) . Assume the population continues to change linearly. a. Find a formula for the moose population, \(P\) since 1990 . b. What does your model predict the moose population to be in 2003\(?\)

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years: $$(2500,2000),(2650,2001),(3000,2003),(3500,2006),(4200,2010)$$ Use linear regression to determine a function \(y\), where the year depends on the population. Round to three decimal places of accuracy.

For the two linear functions, find the point of intersection: \(x=y+2\) \(2 x-3 y=-1\)
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