Problem 37 Use a graphing device to graph t... [FREE SOLUTION]

91影视

Precalculus: Mathematical for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 1: Problem 37

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? $$y=-2 x+b \quad \text { for } b=0, \pm 1, \pm 3, \pm 6$$

Short Answer

Expert verified

The lines are all parallel; they share the same slope of -2.

Step by step solution

Understand the equation

The given equation is a family of lines represented by the equation $y = -2x + b$, where $b$ is a constant that changes. The slope of the line is $-2$ and represents how steep the line is. The constant $b$ is the y-intercept, where the line crosses the y-axis. We will graph these lines for various values of $b$.

List values of b

We need to graph the equation for $b = 0, \pm 1, \pm 3, \pm 6$. This means we will create seven lines in total, each corresponding to a different value of $b$.

Graph the lines

Using a graphing device or graphing software, plot each line by substituting the value of $b$ into the equation $y = -2x + b$. This will produce seven lines which are parallel to each other because they all have the same slope, $-2$.

Interpret what the lines have in common

All lines in this family will be parallel. This is because they share the same slope of $-2$. The only factor that differentiates them is the y-intercept, $b$, which shifts the line up or down along the y-axis. Parallel lines never intersect, so these lines will not cross each other on the graph.

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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 of a linear equation is a way of expressing a straight line's formula in math. It's written as $y = mx + b$. Here, $m$ is the slope, which tells us how steep the line is, and $b$ is the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful in graphing because it easily shows both the slope and the y-intercept.

**Slope (m)** - This is a measure of the line's steepness. A positive slope means the line rises as it moves from left to right, while a negative slope means it descends.
**Y-Intercept (b)** - This indicates the point on the y-axis where the line intersects. It effectively gives the starting point for the line on the vertical axis.

In general, the slope-intercept form directly shows the direction and position of the line, making it easy to graph.

Parallel Lines

Parallel lines are lines in a plane that never meet. They are always the same distance apart and have the same slope. In the context of graphing linear equations in slope-intercept form, if two lines have the same slope, they are parallel.
For instance, consider the equation $y = -2x + b$. No matter what value $b$ takes, the slope $-2$ remains the same. Therefore, changing $b$ results in different lines: $y = -2x + 0$, $y = -2x + 1$, $y = -2x - 1$, etc. All these lines have the same slope, $-2$, and thus are parallel.

Parallel lines share a common slope.
They never intersect.
The y-intercept changes the vertical position without altering slope.

Understanding parallel lines helps in identifying family lines on a graph and ensuring you're working with the right lines.

Y-Intercept

The y-intercept is a fundamental component of the slope-intercept form. It's the value of $b$ in the equation $y = mx + b$. The y-intercept tells us where the line crosses the y-axis. For the equation $y = -2x + b$, the y-intercept varies depending on the value of $b$.

If $b = 0$, the line crosses the y-axis at the origin (0,0).
If $b = 1$, then the line crosses the y-axis one unit above the origin (0,1).
A negative $b$ means crossing below the origin.

Changing $b$ shifts the line up or down the y-axis without changing its slope. Therefore, the y-intercept is a critical factor for determining where a line sits verticially on a graph. It essentially provides the initial position of the line on the graph.

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91影视

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? $$y=-2 x+b \quad \text { for } b=0, \pm 1, \pm 3, \pm 6$$

Short Answer

Step by step solution

Understand the equation

List values of b

Graph the lines

Interpret what the lines have in common

Key Concepts

Slope-Intercept Form

Parallel Lines

Y-Intercept

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