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Chapter 3: Problem 38

Use intercepts and a checkpoint to graph equation. \(y-4 x=0\)

Short Answer

Expert verified
The intercept of the line is (0,0) and the checkpoint (1,1) is not on the line. The graph is a straight line passing through the origin (0, 0)

Step by step solution

01

Find the x-intercept

The x-intercept is the point where the line crosses the x-axis. At this point, the y value is 0. This gives the equation 0 - 4x = 0. Solving for x, getting \(x = 0\)
02

Find the y-intercept

The y-intercept is where the line crosses the y-axis. At this point, the x value is 0. This gives the equation y - 4*0 = 0. Solving for y, getting \(y = 0\)
03

Find a checkpoint

Choose a point not on the line as a checkpoint, For example, choose (1,1). Then, plug these values into the equation i.e. 1 - 4*1=0, which is not true. Hence, (1,1) is not on the line.
04

Graph the equation

Now, plot the points (0,0) which is the x-intercept and the y-intercept on a graph, making sure the put the x-intercept on the x-axis and the y-intercept on the y-axis. The line passing through these points is the graph of the equation.

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Key Concepts

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

x-intercept
The x-intercept of a linear equation is an essential point where the line touches the x-axis. To find this intercept for any equation, you set the value of \( y \) to zero and solve for \( x \). This tells us exactly where the line passes the x-axis.
For the given equation \( y - 4x = 0 \), we substitute \( y = 0 \), which simplifies to:
  • \( 0 - 4x = 0 \)
  • Solve for \( x \): \( 4x = 0 \)
  • \( x = 0 \)
This means that the x-intercept is at the origin, the point \((0, 0)\). Knowing the x-intercept helps us quickly mark a point on the x-axis when graphing the line, providing a starting reference.
y-intercept
The y-intercept is another key point where the line intersects the y-axis. To find it, set \( x = 0 \) in the equation, and solve for \( y \). This approach helps us find exactly where the line hits the y-axis, making it possible to draw accurate graphs.
Considering our equation \( y - 4x = 0 \), substitute \( x = 0 \), which simplifies to:
  • \( y - 4*0 = 0 \)
  • Simplifies to \( y = 0 \)
So here, the y-intercept is also \( (0,0) \), indicating that this line travels through the origin. The y-intercept is useful for plotting points as it shows how high the graph touches or crosses the y-axis.
algebraic checkpoints
An algebraic checkpoint is a strategic method to confirm a point does not lie on the graph. By picking a random point, substituting its values into the equation, and verifying whether the equation holds true, you can ensure accuracy of plotting.
In our example, let's pick the point \((1, 1)\). Substituting into the equation, we have:
  • \( 1 - 4*1 = 0 \)
  • Simplifies to \( 1 - 4 = -3 \)
As it turns out, \(-3 eq 0\), confirming that \((1,1)\) is not on the line. This check further verifies that the graph is correct.
While optional, using a checkpoint is good practice, especially when first learning to graph equations, as it prevents mistakes and enhances understanding of graph behavior.

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Most popular questions from this chapter

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