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Chapter 4: Problem 13

Graph each equation by finding the intercepts and at least one other point. $$y=-2 x+6$$

Short Answer

Expert verified
The y-intercept is (0, 6), the x-intercept is (3, 0), and an additional point is (1, 4). Plot these points on the graph and draw a straight line passing through them to represent the linear equation \(y = -2x + 6\).

Step by step solution

01

Finding the y-intercept

To find the y-intercept, set x to 0 and solve for y. This would give you the point where the graph intersects the y-axis. For the given equation, when x = 0: \(y = -2 (0) + 6 = 6\) The y-intercept is the point (0, 6).
02

Finding the x-intercept

To find the x-intercept, set y to 0 and solve for x. This would give you the point where the graph intersects the x-axis. For the given equation, when y = 0: \(0 = -2x + 6\) Solve for x: \(2x = 6\) \(x = 3\) The x-intercept is the point (3, 0).
03

Finding one more point

To find another point on the graph, choose a value for x and solve for y. This would give you an additional point on the graph. Let x = 1: \(y = -2 (1) + 6 = 4\) So, one more point on the graph is (1, 4). Now, plot the three points (0, 6), (3, 0), and (1, 4) on the graph, and draw a straight line passing through these points to represent the linear equation \(y = -2x + 6\).

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Key Concepts

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

Finding intercepts
Finding intercepts is a fundamental concept when graphing linear equations. The intercepts provide important clues about the points where the graph crosses the axes.

To find the **y-intercept**, we set the x-value to zero and solve for y. This gives us the point at which the line intersects the y-axis.
  • In our example, the equation is \( y = -2x + 6 \). By setting \( x = 0 \), we get \( y = 6 \), making the y-intercept (0, 6).
The **x-intercept** is found by setting the y-value to zero and solving for x. This yields the point where the line intersects the x-axis.
  • Substitute \( y = 0 \) in the equation to find \( x \): \( 0 = -2x + 6 \), solving gives \( x = 3 \). Thus, the x-intercept is (3, 0).
Identifying these intercepts gives us crucial points needed for plotting the line accurately.
Solving linear equations
Solving linear equations involves manipulating the equation to find values of unknown variables. This skill is essential for graphing, as it helps in determining intercepts and other points on the line.

A linear equation like \( y = -2x + 6 \) is clear-cut. To solve for y, you simply replace x with any value you're interested in evaluating.
  • For instance, finding the x-intercept requires setting y to zero and solving: \( 0 = -2x + 6 \). Rearrange to get \( 2x = 6 \), and divide by 2 to find \( x = 3 \).
This straightforward approach is what makes linear equations manageable and easy to work with when graphing.
Plotting points
Plotting points on a graph is a way to visually represent the solutions of a linear equation. Once the intercepts and another point are found, plotting them accurately is critical.

To plot, we use the coordinates: the first number representing the x-value and the second number, the y-value.
  • Begin with the intercepts: Place (0, 6) and (3, 0) on your graph.
  • Add the third point for validation, such as (1, 4) in our example.
Ensure that each point is placed precisely on the graph. Then, connect these points with a straight line by using a ruler for accuracy. This forms the graph of the linear equation \( y = -2x + 6 \).
Graph interpretation
Interpreting the graph of a linear equation is about understanding what the visual representation conveys. The plotted line provides insights into the equation's behavior and its real-world implications.

By observing the slope and intercepts on the graph, we can determine how the line moves.
  • The rightward slant of the line from top to bottom demonstrates a **negative slope**. This is a characteristic of the equation \( y = -2x + 6 \).
  • Intercept points such as (0, 6) and (3, 0) confirm where the graph crosses the y-axis and x-axis, respectively.
The entire process, from finding points to interpreting the graph, links our understanding of equations to visual learning, making abstract concepts tangible and easier to comprehend.

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Most popular questions from this chapter

A \(16 \times\) DVD recorder can transfer \(21.13 \mathrm{MB}\) (megabytes) of data per second onto a recordable DVD. The function \(D(t)=21.13 t\) describes how much data, \(D\) (in megabytes), is recorded on a DVD in \(t\) sec. (Source: www.osta.org) a) How much data is recorded after 12 sec? b) How much data is recorded after 1 min? c) How long would it take to record \(422.6 \mathrm{MB}\) of data? d) Graph the function.

Horton's Party Supplies rents a "moon jump" for \(\$ 100\) plus \(\$ 20\) per hour. This can be described by the equation y=20 x+100where \(x\) represents the number of hours and \(y\) represents the cost. a) Complete the table of values, and write the information as ordered pairs. (table cannot copy) b) Label a coordinate system, choose an appropriate scale, and graph the ordered pairs. c) Explain the meaning of the ordered pair \((4,180)\) in the context of the problem. d) Look at the graph. Is there a pattern indicated by the points? e) For how many hours could a customer rent the moon jump if she had \(\$ 280 ?\)

Graph each function using the slope and \(y\) -intercept. $$g(x)=3 x+3$$

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Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. parallel to \(-3 x+4 y=8\) containing \((7,4)\)
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