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Chapter 3: Problem 87

Use slope-intercept graphing to graph the equation. $$ y=-4 x+3 $$

Short Answer

Expert verified
Plot the y-intercept (0, 3), use the slope -4 to find (1, -1), and draw the line through these points.

Step by step solution

01

Identify the slope and y-intercept

The given equation is in the slope-intercept form, which is:\[ y = mx + b \]Here, the slope (\( m \)) is -4, and the y-intercept (\( b \)) is 3.
02

Plot the y-intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. For \( b = 3 \), plot the point (0, 3) on the graph.
03

Use the slope to find another point

The slope is -4, which means that for every 1 unit you move to the right on the x-axis, you move 4 units down on the y-axis. Starting from the point (0, 3), move 1 unit to the right (x = 1) and 4 units down to get the point (1, -1).
04

Draw the line through the points

With the points (0, 3) and (1, -1) plotted, draw a straight line through these points. This line represents the graph of the equation \( y = -4x + 3 \).

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Key Concepts

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

headline of the respective core concept
Graphing linear equations is a fundamental skill in algebra. It's all about picturing equations as lines on a coordinate plane. When we talk about linear equations, we refer to first-degree equations where the highest exponent of the variable is one. The slope-intercept form, which is the most common way to write these equations, is \[ y = mx + b \]. This format makes it easy to identify the slope and the y-intercept, two key components necessary for graphing the equation.
headline of the respective core concept
The slope of a line, represented by \( m \), shows how steep the line is. It tells us the rate of increase or decrease of the y-value as we move along the x-axis. Specifically, the slope is calculated as the change in y divided by the change in x, often described as 'rise over run.' For example, a slope of -4 means that for every 1 unit move to the right on the x-axis, the line goes down 4 units. A positive slope indicates an upward trend, whereas a negative slope shows a downward trend.
headline of the respective core concept
The y-intercept, indicated by \( b \), is the point where the line crosses the y-axis. In the equation \[ y = -4x + 3 \], the y-intercept is 3. To locate it on the graph, you plot the point where \( x = 0 \), resulting in the coordinate (0, 3). This point is crucial because it helps you to start plotting the line. Combined with another point derived using the slope, you can draw the line representing the equation. This way, you visualize how the equation formulates a straight path across the graph.

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Most popular questions from this chapter

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The number of traffic fatalities in Kentucky decreased from 985 deaths in 2005 to 760 deaths in 2010 . (Source: www-nrd.nhtsa.dot.gov)

Explain why the \(x\)-intercept, the \(y\)-intercept, and one other point are not three different solutions of \(y=2 x\).

For problems 99-102, solve. \(m=\frac{10-4}{5-3}\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((1,5) ;(3,13)\)

For exercises 67-78, (a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(4 x+9 y=72\)
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