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Chapter 2: Problem 56

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ x-4=0 $$

Short Answer

Expert verified
The x-intercept is at (4, 0), and there is no y-intercept. The graph is a vertical line at x = 4.

Step by step solution

01

- Find the x-intercept

The x-intercept occurs where the graph crosses the x-axis, which means that the y-value is 0. The given equation is: \[ x - 4 = 0 \] To find the x-intercept, solve for x: \[ x - 4 = 0 \] Add 4 to both sides to isolate x: \[ x = 4 \] So, the x-intercept is at \( (4, 0) \).
02

- Find the y-intercept

The y-intercept occurs where the graph crosses the y-axis, which means that the x-value is 0. However, for the equation \( x - 4 = 0 \), x is always 4, and there is no y component in this equation. This means the line is vertical and does not cross the y-axis. So, there is no y-intercept.
03

- Graph the equation

To graph the equation \( x - 4 = 0 \), plot the x-intercept at \( (4, 0) \). Since this is a vertical line, draw a straight line through the point \( (4, 0) \) that goes up and down, parallel to the y-axis.

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Key Concepts

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

Graphing Equations
Graphing equations is a fundamental skill in algebra that helps visualize mathematical relationships. To graph an equation, we need to identify key points where the graph intersects the axes and plot them on the coordinate plane. This process often involves finding the intercepts, which are the points where the graph crosses the x-axis and y-axis.

Once you find these intercepts, plotting them creates a framework for the entire graph. For linear equations, this usually means drawing a straight line through the points. Remember, accuracy in plotting these points ensures that your graph represents the correct equation.
Vertical Lines
Vertical lines occur when an equation is in the form of \( x = k \). This means that x is always a constant value, while y can take any value. In this particular exercise, the equation \( x - 4 = 0 \) simplifies to \( x = 4 \), indicating a vertical line where x is always 4.

Vertical lines are unique because they do not have a y-intercept; they run parallel to the y-axis. To graph a vertical line, you simply plot the x-intercept and draw a straight line up and down through that point, which remains parallel to the y-axis. This line extends infinitely in both directions.
Solving Linear Equations
Solving linear equations is crucial for finding intercepts and graphing lines. For equations involving one variable, like \( x - 4 = 0 \), solving it involves isolating the variable on one side of the equation.

Here is how you solve \( x - 4 = 0 \) to find the x-intercept:
  • Add 4 to both sides to isolate x: \( x = 4 \)
  • So, the x-intercept is at (4, 0)
Mastering this method is essential for handling more complex linear equations in the future.
Intercepts in Coordinate Plane
Intercepts are vital points on the graph of an equation. The x-intercept is the point where the graph crosses the x-axis, and the y-intercept is where it crosses the y-axis. For linear equations, these intercepts help define the line's position and direction.

To find intercepts:
  • X-intercept: Set y to 0 and solve for x.
  • Y-intercept: Set x to 0 and solve for y.
In this exercise, since we have \( x - 4 = 0 \), we determined the x-intercept to be (4, 0), but there is no y-intercept as the line is vertical. Understanding intercepts helps to accurately graph and comprehend the behavior of equations on a coordinate plane.

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Most popular questions from this chapter

Graph each line passing through the given point and having the given slope. (0,-4)\(; m=-\frac{3}{2}\)

Graph each linear or constant function. Give the domain and range. $$ f(x)=0 $$

For each function, find (a) \(f(2)\) and (b) \(f(-1)\) $$ f=\\{(2,5),(3,9),(-1,11),(5,3)\\} $$

If the graph of a linear equation rises from left to right, then the average rate of change is (positive / negative). If the graph of a linear equation falls from left to right, then the average rate of change is (positive / negative).

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