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Chapter 3: Problem 38

Graph each linear equation. \(y=4 x+3\)

Short Answer

Expert verified
Plot points (0,3) and (1,7), then draw a line through them.

Step by step solution

01

Understand the Equation

The given equation is in slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept.
02

Identify the Slope and Y-intercept

In the equation \(y=4x+3\), the slope \(m\) is 4, and the y-intercept \(b\) is 3. This means the line will cross the y-axis at (0, 3).
03

Plot the Y-intercept

Start by plotting the y-intercept (0, 3) on the graph. This is the point where the line crosses the y-axis.
04

Use the Slope to Find Another Point

The slope of 4 means that for every 1 unit increase in x, y increases by 4 units. From the y-intercept (0, 3), move 1 unit to the right (positive x direction) and 4 units up (positive y direction) to find the point (1, 7).
05

Plot the Second Point

Plot the second point (1, 7) on the graph. This gives a second point through which the line will pass.
06

Draw the Line

Using a ruler, draw a straight line through the points (0, 3) and (1, 7) extending in both directions. This is 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.

slope-intercept form
Linear equations are often expressed in the slope-intercept form, which is written as: y = mx + bThis is a very useful form because it clearly shows two important features of the line: the slope (m) and the y-intercept (b). In the given equation y = 4x + 3, the slope (m) is 4, and the y-intercept (b) is 3.The slope-intercept form helps us quickly identify how steep the line is and where it crosses the y-axis. This form is essential for graphing linear equations easily.
slope
The slope of a line indicates how steep the line is and the direction it goes.It is represented by the letter m. For the equation y = 4x + 3, the slope is 4.The slope tells us that for every one unit increase in the x-direction (rightward), the value of y (upward) increases by 4 units.If the slope is positive, the line rises as it moves from left to right. If it's negative, the line falls as it moves from left to right. In this case, a slope of 4 means the line rises steeply.
y-intercept
The y-intercept is where the line crosses the y-axis and is represented by the letter b. For the equation y = 4x + 3, the y-intercept (b) is 3.This means the line will pass through the point (0, 3) on the graph. To find this point, you set x to 0 and solve for y. For example: y = 4(0) + 3 = 3.This point gives us a starting position to begin graphing our line. From this point, you use the slope to find other points.
plotting points
To graph a linear equation, you need at least two points. The simplest point to plot is often the y-intercept, which we already determined is (0, 3).Next, use the slope to find another point. With a slope of 4, from the y-intercept (0, 3), move one unit to the right (increase x by 1) and four units up (increase y by 4). This gets you to the point (1, 7).
Plot both points on graph paper, and then use a ruler to draw a straight line through these points. Extend the line in both directions. This line represents the graph of the equation y = 4x + 3.

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Most popular questions from this chapter

For each function \(f,\) find \((a) f(2),(b) f(0),\) and \((c) f(-3) .\) See Example 5 $$ f(x)=x^{2}-x+2 $$

A point \((x, y)\) has the property that \(x y>0 .\) In which quadrant(s) must the point lie? Explain.

Find the \(x\) -intercept and the \(y\) -intercept for the graph of each equation. $$ y=\frac{1}{3} x-2 $$

Plot each set of points, and draw a line through them. Then give the equation of the line. $$ (3,5),(3,0), \text { and }(3,-3) $$

For each pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither. See Example \(6 .\) $$ \begin{aligned} &3 x-5 y=-1\\\ &5 x+3 y=2 \end{aligned} $$
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