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Chapter 3: Problem 16

Find the \(x\) - and \(y\) -intercepts. $$ y=-3 $$

Short Answer

Expert verified
The \( y \)-intercept is (0, -3); there is no \( x \)-intercept.

Step by step solution

01

Identify the Type of Equation

The given equation is a horizontal line, represented as \( y = -3 \). This means the value of \( y \) is always -3, regardless of \( x \).
02

Find the y-intercept

The \( y \)-intercept is the point where the line crosses the \( y \)-axis. For horizontal lines, the \( y \)-intercept is simply \( y = -3 \). Therefore, the \( y \)-intercept is the point \((0, -3)\).
03

Find the x-intercept

The \( x \)-intercept is the point where the line crosses the \( x \)-axis. Since this is a horizontal line with a constant \( y \) value of -3, it never crosses the \( x \)-axis. Therefore, there is no \( x \)-intercept.

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Key Concepts

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

Horizontal Line
A horizontal line is a straight line where all points have the same y-coordinate. The equation of a horizontal line is in the form of \( y = c \), where \( c \) is a constant. This means no matter what value \( x \) takes, \( y \) remains unchanged. In our example, \( y = -3 \), so the line is parallel to the x-axis and all points on it have a y-value of -3.
  • Horizontal lines always run left to right across the graph.
  • The slope of a horizontal line is 0, indicating no vertical change.
  • Such lines never cross the x-axis unless \( y = 0 \).
x-intercept
The x-intercept is where a line crosses the x-axis. For any point on the x-axis, the y-coordinate is 0. To find the x-intercept, you set \( y = 0 \) in the equation and solve for \( x \).
For example, if we try to find the x-intercept for the line \( y = -3 \), setting \( y = 0 \) doesn't work because the equation has no solution where y equals zero. Therefore:
  • Horizontal lines like \( y = -3 \) have no x-intercept.
  • They are parallel to the x-axis and never touch it.
It's helpful to remember that only non-horizontal lines will typically have x-intercepts.
y-intercept
The y-intercept is found where a line crosses the y-axis. For horizontal lines, it's particularly simple to determine. Since the y-value is constant across the entire line, the y-intercept is just that value.
For the line \( y = -3 \):
  • The y-intercept is \( (0, -3) \).
  • This is because at \( x = 0 \), the y-value remains at -3.
Understanding y-intercepts is crucial, as it gives insight into where the line sits vertically on the graph.
Linear Equations
Linear equations represent straight lines on a graph and can typically be described by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
The equation \( y = -3 \) is a simplified linear equation:
  • It has no \( x \)-term, indicating a zero slope.
  • Its form is consistent with all y-values being the same.
This highlights the nature of linear equations where simplicity in format often provides clear information about the graph's characteristics.

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