Problem 12 - Precalculus with Limits

Question

In Exercises 11 and 12, sketch the lines through the point with the indicated slopes on the same set of coordinate axes. Point \( (-4, 1) \) Slopes (a) \(3\) (b) \(-3\) (c) \(\frac{1}{2}\) (d) Undefined

Step-by-Step Solution

Answer

The lines are respectively upwards tilted to the right for a slope of 3, downwards tilted to the right for a slope of -3, slightly upwards tilted to the right for a slope of \(\frac{1}{2}\), and vertical for an undefined slope

Step 1: Graph the Line with slope 3

Start at the given point \((-4, 1)\). Since the slope is 3, which means for every run of 1, the rise is 3. From the point \((-4, 1)\), move 1 unit to the right(horizontally), and 3 units up (vertically), plot this point. Connect the given point \((-4, 1)\) and the new point, this gives the line with slope 3.

Step 2: Graph the Line with slope -3

Start at the given point \((-4, 1)\). The slope -3 means for every run of 1, the rise is -3 or for every run of -1, the rise is 3. From the point \((-4, 1)\), one can either move 1 unit to the right and 3 units down, or move 1 unit to the left and 3 units up, plot this new point. Connect the given point \((-4, 1)\) and the new point, this gives the line with slope -3.

Step 3: Graph the Line with slope \(\frac{1}{2}\)

Start at the given point \((-4, 1)\). The slope \(\frac{1}{2}\) means that for every run of 2, the rise is 1. From the point \((-4, 1)\), move 2 units to the right and 1 unit up, plot this new point. Connect the given point \((-4, 1)\) and the new point, this gives the line with slope \(\frac{1}{2}\).

Step 4: Graph the Line with Undefined Slope

An undefined slope implies the line is vertical. Draw a vertical line that passes through the given point \((-4, 1)\)

Key Concepts

Slopes

Slopes are a fundamental concept in graphing linear equations. The slope of a line is a measure of its steepness and direction. It's determined by the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.

• If a slope is positive, the line rises from left to right.
• If a slope is negative, the line falls from left to right.
• A slope of zero indicates a horizontal line.
• An undefined slope signifies a vertical line.

In this exercise, the slopes provided are positive (\(3\)), negative (\(-3\)), fractional (\(\frac{1}{2}\)), and undefined. By understanding slopes, you have a tool to predict and graph the direction and angle of a line on a plane. This concept is crucial to solving many types of algebraic problems.

Coordinate Axes

The coordinate axes are the foundation for graphing any point or line on a plane. These axes consist of two perpendicular lines: the horizontal x-axis and the vertical y-axis. Together, they create a grid used to plot equations.

• The x-axis runs horizontally and measures left and right from the origin (0,0).
• The y-axis runs vertically and measures up and down from the origin.

Every point on the plane is defined by an ordered pair \((x, y)\), indicating its position relative to these axes. Understanding the coordinate axes is essential for interpreting and graphing linear equations. It helps in locating points and understanding the movement dictated by the slope of a line.

Points on a Plane

Graphing equations involves plotting points on a plane. A point is described by an \((x, y)\) coordinate that places it in relation to the x-axis and y-axis.

• The x-value tells you how far left or right to move from the origin.
• The y-value indicates how far up or down to move from the origin.

For instance, the point \((-4, 1)\) means starting at the origin, moving 4 units left (as \(-4\) is negative), and then 1 unit up. In graphing linear equations, starting at a specific point and applying the slope lets you draw the line that represents the equation. Each point along the line confirms it follows the slope and orientation you expect.

Vertical Lines

Vertical lines are a special type of line in graphing linear equations. They are characterized by an undefined slope. This is because a vertical line rises infinitely without running horizontally, making the calculation of slope division by zero (which is undefined).

• Vertical lines are parallel to the y-axis.
• Every point on the line has the same x-coordinate.

In this exercise, the line with an undefined slope through the point \((-4, 1)\) simply means that the entire line consists of points that have \(x = -4\). These lines visually emphasize the importance of understanding how slopes and axes interact, even when traditional slope calculations don't apply.