Problem 28
Question
Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). Do not use a calculator. 2\(x=-3\)
Step-by-Step Solution
Verified Answer
(a) The x-intercept is \((-\frac{3}{2}, 0)\). (b) No y-intercept exists. (c) Domain: \{x = -\frac{3}{2}\}. (d) Range: all real numbers. (e) The slope is undefined.
1Step 1: Rewrite the Equation
The given equation is \(2x = -3\). To clearly work with this equation, rewrite it in the standard linear form \(Ax + By = C\). Set \(B = 0\) to get \(2x + 0 \cdot y = -3\), meaning this is a vertical line.
2Step 2: Identify the x-intercept
For vertical lines, the line intersects the x-axis directly at whatever x-value satisfies the equation. Solving \(2x = -3\), we get \(x = -\frac{3}{2}\). Thus, the x-intercept is \((-\frac{3}{2}, 0)\).
3Step 3: Determine the y-intercept
Since the equation represents a vertical line, it does not cross the y-axis at any point. Therefore, there is no y-intercept.
4Step 4: Determine the Domain
Vertical lines have the same x-value for all points on the line and no restrictions on y. Therefore, the domain is a single value: \(x = -\frac{3}{2}\).
5Step 5: Determine the Range
Because the vertical line extends infinitely up and down on the y-axis, the range is all real numbers, \((-\infty, +\infty)\).
6Step 6: Identify the Slope
A vertical line does not have a defined slope. In terms of calculus, the slope would be considered undefined or infinite, as vertical lines have undefined slope.
Key Concepts
Understanding x-interceptsDecoding the y-interceptExplaining domain and rangeUnderstanding the undefined slope
Understanding x-intercepts
For any line on a graph, the x-intercept refers to the point where the line crosses the x-axis. This is where the y-value is zero. In simpler terms, it's the value of x when y equals zero.
When you have an equation like a vertical line, the x-intercept is straightforward to find. Consider the equation given: \(2x = -3\).
To solve for the x-intercept, you set y to zero and solve for x. This means you solve the equation \(2x = -3\) to identify the x-value. Here, the x-intercept is found to be at \((-\frac{3}{2}, 0)\).
Remember that vertical lines might only have one x-intercept, differing from other types of lines.
When you have an equation like a vertical line, the x-intercept is straightforward to find. Consider the equation given: \(2x = -3\).
To solve for the x-intercept, you set y to zero and solve for x. This means you solve the equation \(2x = -3\) to identify the x-value. Here, the x-intercept is found to be at \((-\frac{3}{2}, 0)\).
Remember that vertical lines might only have one x-intercept, differing from other types of lines.
Decoding the y-intercept
Unlike the x-intercept, where the line crosses the x-axis, the y-intercept is where the line crosses the y-axis. This is located at the point where x is zero.
With the equation \(2x = -3\), which represents a vertical line, notice something different.
Vertical lines are peculiar because they only have a specific x-value, meaning they typically don't meet the y-axis at all.
As such, they have no y-intercept. This characteristic makes vertical lines quite unique compared to diagonal or horizontal lines that often have both x and y-intercepts.
With the equation \(2x = -3\), which represents a vertical line, notice something different.
Vertical lines are peculiar because they only have a specific x-value, meaning they typically don't meet the y-axis at all.
As such, they have no y-intercept. This characteristic makes vertical lines quite unique compared to diagonal or horizontal lines that often have both x and y-intercepts.
Explaining domain and range
The domain and range concepts involve understanding which x and y values a function (or line) can occupy.
For any line, the domain refers to all possible x values. A vertical line's domain is simply one x-value. In our case, with the equation \(2x = -3\), the domain is exclusively \(x = -\frac{3}{2}\).
The range, on the other hand, is about all potential y-values the line can take. Since vertical lines extend forever up and down, without any horizontal movement, the range is all real numbers. You can think of it as \((-\infty, +\infty)\).
Remember: domain is about horizontal span (x-values), while range is about vertical reach (y-values).
For any line, the domain refers to all possible x values. A vertical line's domain is simply one x-value. In our case, with the equation \(2x = -3\), the domain is exclusively \(x = -\frac{3}{2}\).
The range, on the other hand, is about all potential y-values the line can take. Since vertical lines extend forever up and down, without any horizontal movement, the range is all real numbers. You can think of it as \((-\infty, +\infty)\).
Remember: domain is about horizontal span (x-values), while range is about vertical reach (y-values).
Understanding the undefined slope
When discussing the slope of a line, it's the measure of how steep or slanted the line is. Generally, slope is calculated as "rise over run," or the change in y over the change in x.
However, vertical lines, like the one described by \(2x = -3\), do not have a defined slope.
Why? Because the rise (y) happens without any run (x), making the run zero. When you try to divide any number by zero, it becomes undefined. This is why vertical lines have what's called an "undefined slope."
A practical way to remember: if a line goes straight up and down, it has an undefined slope, distinguishing it from lines that have calculable slopes.
However, vertical lines, like the one described by \(2x = -3\), do not have a defined slope.
Why? Because the rise (y) happens without any run (x), making the run zero. When you try to divide any number by zero, it becomes undefined. This is why vertical lines have what's called an "undefined slope."
A practical way to remember: if a line goes straight up and down, it has an undefined slope, distinguishing it from lines that have calculable slopes.
Other exercises in this chapter
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