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Chapter 2: Problem 4

Find the slope of the graph of each equation, if possible. a. \(y=-x\) b. \(x=-3\)

Short Answer

Expert verified
a. Slope is -1; b. Slope is undefined.

Step by step solution

01

Identify the form of the equation

The equation of a line can be written in slope-intercept form as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Let's identify if the equations given fit this form.
02

Analyze equation a: \(y = -x\)

This equation is already in the slope-intercept form, \(y = mx + b\). We can directly see that \(m = -1\). Hence, the slope of the equation \(y = -x\) is -1.
03

Analyze equation b: \(x = -3\)

This equation represents a vertical line, because it is of the form \(x = c\), where \(c\) is a constant. Vertical lines have an undefined slope because the change in y can be any value while the change in x is zero, which is division by zero.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a popular way to express the equation of a line. This form is written as \( y = mx + b \). Here, \(m\) represents the slope of the line, while \(b\) indicates the y-intercept. The y-intercept is the point where the line crosses the y-axis.
  • The slope \(m\) indicates the steepness and direction of the line. If \(m\) is positive, the line slants upwards. If negative, it slants downwards.
  • The y-intercept \(b\) shows the starting point of the line on the y-axis when \(x = 0\).
If you have an equation like \(y = -x\), it is already in slope-intercept form, where \(m = -1\) and \(b = 0\). So, the slope of this line is -1, meaning it goes downwards, passing through the origin at (0, 0).
Understanding this form makes it easier to draw the line on a graph and interpret its characteristics.
Vertical Line
A vertical line is a special type of line that runs straight up and down. This line is always described by an equation of the form \(x = c\), where \(c\) is a constant value.
  • In the case of equation \(x = -3\), this means that the line crosses the x-axis at -3 and moves vertically.
  • Vertical lines are unique because they do not have a traditional \(y = mx + b\) form.
What鈥檚 fascinating about vertical lines is that they have consistent x-values but variable y-values, portraying a constant x-position as y changes. This makes them distinct in terms of slope.
Undefined Slope
When discussing slopes, we often refer to them as numbers that describe how steep a line is. With a vertical line, however, calculating the slope becomes tricky.
  • Slope is calculated as \( \frac{\text{change in y}}{\text{change in x}} \).
  • In vertical lines, the change in x is 0.
  • Since dividing by zero is mathematically undefined, vertical lines are said to have an undefined slope.
So, for the line \(x = -3\), while the y-values may change, the x remains constant, leading to an undefined slope. This means you can't assign a numerical value to describe their steepness like with non-vertical lines.

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