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Chapter 11: Problem 264

In the following exercises, graph the line given a point and the slope. $$ (-4,2) ; m=4 $$

Short Answer

Expert verified
Equation: \( y = 4x + 18 \); Plot points \((-4, 2)\) and \((-3, 6)\); Draw line.

Step by step solution

01

Understand the Point-Slope Formula

The point-slope formula for the equation of a line is given by \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope of the line.
02

Substitute Point and Slope into the Formula

Substitute the given point \( (-4, 2) \) and the given slope \( m = 4 \) into the point-slope form of the equation: \[ y - 2 = 4(x + 4) \]
03

Simplify the Equation

Distribute the slope and simplify the equation: \[ y - 2 = 4x + 16 \] \[ y = 4x + 18 \]
04

Plot the Point

On a graph, plot the given point \((-4, 2)\).
05

Use the Slope to Find Another Point

From the point \((-4, 2)\), use the slope \( m = 4 \). Since the slope means 'rise over run', go up 4 units and 1 unit to the right to find another point, \((-3, 6)\).
06

Draw the Line

Draw a straight line through the points \((-4, 2)\) and \((-3, 6)\). This is the graph of the equation \( y = 4x + 18 \).

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Key Concepts

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

Point-Slope Formula
The point-slope formula is key to writing the equation of a line when you know a point on the line and its slope. This formula is expressed as \[ y - y_1 = m(x - x_1) \]. Here, \( (x_1, y_1) \) represents any point on the line, and \( m \) is the slope. It allows you to capture all possible y-values based on the x-values and the given slope.
Suppose we know that a line passes through the point \( (-4, 2) \) and has a slope \( m = 4 \). Plugging these values into the point-slope formula, we get: \[ y - 2 = 4(x + 4) \]. This represents the equation of our line.
Plotting Points
Plotting points is integral to graphing linear equations. A point is denoted as \( (x, y) \) where \( x \) indicates the horizontal position and \( y \) the vertical position on a graph.
First, plot the given point \( (-4, 2) \). Start at the origin \( (0, 0) \), then move 4 units to the left for the x-coordinate (-4), and 2 units up for the y-coordinate (2). Mark this point.
Once you have your starting point, use the slope to plot another point.
Calculating Slope
Slope measures the steepness of a line and is calculated as the 'rise over run', which means \( \text{slope} = \frac{\text{change in } y}{\text{change in } x} \). For the given slope \( m = 4 \), this means for every 1 unit you move horizontally to the right, you move 4 units vertically up.
From the point \( (-4, 2) \), move 1 unit right to \( -3 \) and 4 units up to \( 6 \). This gives another point \( (-3, 6) \) on your line.
Finally, use a ruler to draw a straight line through the two points. This line represents the equation \( y = 4x + 18 \).
Understanding how to calculate slope simplifies graphing linear equations and enhances interpreting and predicting data.

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