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Chapter 2: Problem 4

Graph each equation. Check your work. $$ y=-4 x+5 $$

Short Answer

Expert verified
Graph by plotting the y-intercept at (0,5) and using the slope -4 to find a second point at (1,1), then draw a line through these two points.

Step by step solution

01

Identify the slope and y-intercept

For the equation in slope-intercept form, which is given by the equation y = mx + b, identify the slope (m) and the y-intercept (b). In this equation, the slope (m) is -4 and the y-intercept (b) is 5.
02

Plot the y-intercept

Begin the graph by plotting the y-intercept (0, 5) on the y-axis. This is the point where the line will cross the y-axis.
03

Use the slope to find another point

From the y-intercept, use the slope to find another point. The slope of -4 means you should move down 4 units and 1 unit to the right (since slope is rise over run, or in this case fall over run, which is -4/1). Plot this second point.
04

Draw the line

Draw a straight line through the y-intercept and the second point to graph the equation of the line.
05

Check your work

Verify your graph by choosing any x-value, calculating the corresponding y-value using the given equation, and then checking that this (x, y) point lies on the line you've drawn.

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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 an easy way to write linear equations and quickly graph them. It is written as \r\( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. This form is particularly helpful as it gives immediate information about the line's properties: how steep it is and where it crosses the y-axis. For instance, in the equation \r\( y = -4x + 5 \), the slope is -4, and the y-intercept is 5. In practice, this means you can begin graphing at point (0, 5) and then use the slope to find another point along the line.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis. When working with the slope-intercept form \r\( y = mx + b \), the y-intercept is given by \( b \). This particular point has an x-coordinate of 0, making it easy to plot on a graph. In our example equation \r\( y = -4x + 5 \), the y-intercept is 5, so you begin your graph with the point (0, 5). From this starting point, you can continue to use the slope to plot the course of the line as you graph.
Plotting Points
Plotting points is a fundamental skill in graphing linear equations. After identifying your starting point, often the y-intercept, continue plotting by using the slope. For each subsequent point, move vertically based on the slope's numerator (rise or fall) and horizontally based on its denominator (run). In our example with the slope of -4, from the y-intercept, you would move down 4 units (since it's negative) and right 1 unit (this is because the slope is -4/1). By plotting several points and connecting them with a line, you create a visual representation of the equation.
Slope of a Line
The slope of a line represents its steepness and the direction it travels. Calculated as 'rise over run', the slope can be positive or negative. A positive slope ascends from left to right, while a negative slope descends. A slope of 0 indicates a horizontal line, and an undefined slope indicates a vertical line. In the equation \r\( y = -4x + 5 \), the slope is -4, which means for every one unit the line moves horizontally to the right, it will fall four units. This gives the line a sharp decline as it moves to the right.

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