Question

Multiple Choice Choose the correct statement about the graph of the line \(y=-3\) (a) The graph is vertical with \(x\) -intercept -3 . (b) The graph is horizontal with \(y\) -intercept -3 . (c) The graph is vertical with \(y\) -intercept -3 (d) The graph is horizontal with \(x\) -intercept -3 .

Short answer

Option (b) is correct: The graph is horizontal with \(y\)-intercept -3.

Step-by-step solution

Understanding the Equation

The given equation is \(y=-3\). This equation is in the format \(y = c\), which describes a horizontal line.

Identify the Slope

Since there is no \(x\) term in the equation \(y = -3\), the slope of the line is 0. This confirms that the line is horizontal.

Determine the Y-intercept

The equation \(y = -3\) indicates that the line crosses the \(y\)-axis at -3. Therefore, the \(y\)-intercept is -3.

Clarify the X-intercept

A horizontal line parallel to the x-axis does not cross the x-axis at any point, so it has no x-intercept.

Choose the Correct Option

Based on the steps above, the correct statement is: (b) The graph is horizontal with \(y\)-intercept -3.

Key concepts

horizontal line

A horizontal line is a straight line that goes from left to right across the graph. It has the same y-value for any x-value. For example, in the equation \(y = -3\), the y-value is always -3 regardless of the x-value. Horizontal lines run parallel to the x-axis and never intersect it. This means they do not have an x-intercept. Horizontal lines are unique because:

• They have a slope of zero.
• They are defined by their constant y-value.

y-intercept

The y-intercept is the point where a line crosses the y-axis. In the equation \(y = -3\), this point is where the value of y is -3. At this point, the x-value is always zero. For the equation \(y = -3\):

• The y-intercept is -3.
• This can be written as the point (0, -3).

The y-intercept helps in quickly sketching the graph of the line, as it shows exactly where the line meets the y-axis.

slope of the line

The slope of a line measures its steepness and direction. It is often denoted by the letter 'm'. For the equation \(y = -3\), there is no \(x\) term present. This indicates that the slope of the line is zero. A zero slope means the line is perfectly horizontal and flat.
Key points about slopes:

• A positive slope means the line rises as it goes from left to right.
• A negative slope means the line falls as it goes from left to right.
• A zero slope, like \(y = -3\), means the line is horizontal.

x-intercept

The x-intercept is the point where a line crosses the x-axis. For the equation \(y = -3\), the line is horizontal and does not intersect the x-axis. Therefore, it does not have an x-intercept.
Some key points about x-intercepts:

• They are found by setting y to zero and solving for x.
• Horizontal lines parallel to the x-axis do not have x-intercepts.
• If an equation has a constant y-value, it won't cross the x-axis.