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Chapter 2: Problem 102

Which of the viewing windows would show both the \(x\) - and \(y\) -intercepts of the graph of \(780 x-42 y=5460 ?\) a. [-20,20,2] by [-40,40,10] b. [-10,10,1] by [-10,10,1] c. [-10,10,1] by [-10,150,10] d. [-10,10,1] by [-150,10,10]

Short Answer

Expert verified
Both options c and d show the intercepts.

Step by step solution

01

Find the x-intercept

The x-intercept occurs when y=0. Substitute 0 for y in the equation and solve for x:\[780x - 42(0) = 5460 \] \[780x = 5460 \] \[x = \frac{5460}{780} = 7\]
02

Find the y-intercept

The y-intercept occurs when x=0. Substitute 0 for x in the equation and solve for y:\[780(0) - 42y = 5460 \] \[-42y = 5460 \] \[y = \frac{-5460}{42} = -130\]
03

Determine the viewing window

Both intercepts should be visible within the window's ranges: x-intercept at x=7 and y-intercept at y=-130.a. [-20,20,2] by [-40,40,10] does not include y=-130.b. [-10,10,1] by [-10,10,1] does not include y=-130.c. [-10,10,1] by [-10,150,10] includes both x=7 and y=-130.d. [-10,10,1] by [-150,10,10] includes both x=7 and y=-130.
04

Select the appropriate viewing window

Options c and d include both intercepts. As both options are correct, you can choose either.

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Key Concepts

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

x-intercept
In graphing linear equations, the x-intercept is the point where the graph crosses the x-axis. At this point, the value of y is zero. To find the x-intercept, substitute 0 for y in the given equation and solve for x. For instance, in the equation 780x - 42y = 5460 when y = 0, the equation simplifies to 780x = 5460 . Solving for x gives: x = 7 . This x-coordinate (7,0) is the x-intercept.
y-intercept
The y-intercept is equally important and is where the graph crosses the y-axis. Here, the x value is zero. To find the y-intercept, set x to 0 in the equation and solve for y. Using the equation 780x - 42y = 5460 , substitute x with 0 to get: -42y = 5460 . Solving for y gives: y = -130 . Thus, the y-coordinate (0, -130) is the y-intercept. This point demonstrates where the line crosses the y-axis.
viewing window
Selecting the correct viewing window is crucial for visualizing the entire graph, especially when plotting the x and y intercepts. A viewing window in graphing defines the range of x and y values displayed on the coordinate system. For the given graph of the equation 780x - 42y = 5460 , the calculated intercepts are at x = 7 and y = -130. Therefore, any appropriate viewing window should at minimum include these points.

The options are:
  • [-20,20,2] by [-40,40,10] - does not include y = -130
  • [-10,10,1] by [-10,10,1] - does not include y = -130
  • [-10,10,1] by [-10,150,10] - includes both intercepts
  • [-10,10,1] by [-150,10,10] - includes both intercepts
Options c and d are correct as they include both intercepts (-10 to 150 on y).
coordinate system
Understanding the coordinate system is fundamental when graphing. The coordinate system is a plane with two perpendicular axes (x and y) that intersect at the origin (0,0). Each point on the plane is determined by a pair of numerical coordinates (x,y):
  • x-coordinate represents horizontal movement along the x-axis
  • y-coordinate represents vertical movement along the y-axis
Both coordinates are essential for graphing linear equations and identifying critical points like intercepts.

In our exercise, knowing the position of (7, 0) and (0, -130) helps determine the correct viewing window that includes all critical points of the graph.

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