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Chapter 3: Problem 25

Find the \(x\) -intercept and the \(y\) -intercept for the graph of each equation. $$ x-4=0 $$

Short Answer

Expert verified
x-intercept: (4, 0); no y-intercept.

Step by step solution

01

Identify the given equation

The given equation is \( x - 4 = 0 \).
02

Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Since y does not appear in the expression, the equation simplifies to \( x - 4 = 0 \). Add 4 to both sides to get \( x = 4 \). So the x-intercept is (4, 0).
03

Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. In this case, the equation \( x = 4 \) is independent of y, so there is no y-intercept. The graph is a vertical line and does not cross the y-axis.

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Key Concepts

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

x-intercept
An x-intercept is where the graph of an equation crosses the x-axis. This means the y-coordinate at this point is always zero.
  • To find it, set y to 0 in the equation and solve for x.
  • For example, in the equation given: \( x - 4 = 0 \).
Since there is no y involved, we simply solve for x:
  • Add 4 to both sides: \( x = 4 \).
So, the x-intercept is at \( (4, 0) \). This means the graph crosses the x-axis at the point where x is 4.
y-intercept
A y-intercept is where the graph crosses the y-axis. At this point, the x-coordinate is always zero.
  • To find it, set x to 0 in the equation and solve for y.
However, in the given equation: \( x - 4 = 0 \),
  • When we set x to 0, we get \( 0 - 4 = 0 \), which is not true.
This tells us the graph doesn't intersect with the y-axis, so there is no y-intercept. This particular equation represents a vertical line, which we鈥檒l explain next.
graphing equations
Graphing can help you visualize solutions and understand the behavior of equations. For the equation \( x - 4 = 0 \), we identified an x-intercept at \( (4, 0) \).
  • We can plot this point right on the graph where the x-axis and 4 meet.
  • Since the equation lacks a y-term, it means x is always constant at 4.
This results in a vertical line that runs straight up and down through x = 4. By graphing this, we can clearly see that:
  • The line does not cross the y-axis, confirming there's no y-intercept.
  • The point \( (4, 0) \) on the x-axis is where our line intersects.
vertical lines
Vertical lines have a unique characteristic in graphing. These lines go straight up and down.
In the equation \( x - 4 = 0 \),
  • The x-value is always 4.
  • This line doesn鈥檛 depend on y, and thus it doesn't tilt or slope.
In fact, any equation of the form \( x = a \), where 'a' is a constant, represents a vertical line. This line will cross the x-axis at 'a' and nowhere else. To summarize, vertical lines:
  • Have no y-intercept.
  • Cross the x-axis exactly once, at the point \( (a, 0) \).
These properties make them easy to understand and plot on a graph!

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