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Chapter 3: Problem 56

Graph by hand. $$y=-\frac{4}{3} x+3$$

Short Answer

Expert verified
Plot the y-intercept (0, 3), use the slope \(m=-\frac{4}{3}\) to find another point (3, -1), 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 written as \(y=mx+b\). Here, \(m\) represents the slope and \(b\) represents the y-intercept. For \(y=-\frac{4}{3}x+3\), the slope \(m=-\frac{4}{3}\) and the y-intercept \(b=3\).
02

- Plot the y-intercept

The y-intercept \(b\) is the point where the line crosses the y-axis. Plot the point \( (0, 3) \) on the graph.
03

- Use the slope to find another point

The slope \(m=-\frac{4}{3}\) tells us how the line moves. Starting from the y-intercept \( (0, 3) \), move down 4 units (since the slope is negative) and right 3 units. Plot this second point at \( (3, -1) \).
04

- Draw the line

Draw a straight line through the points \( (0, 3) \) and \( (3, -1) \). This is the graph of the equation \(y=-\frac{4}{3}x+3\).

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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 a way to write the equation of a straight line. The general formula is given by \[ y = mx + b \]where \( m \) is the slope and \( b \) is the y-intercept. This form is popular because it easily shows the slope and the point where the line crosses the y-axis.

脦n the equation \( y = -\frac{4}{3}x + 3 \), the slope \( m \) is \( -\frac{4}{3} \) and the y-intercept \( b \) is 3. This means the line goes down 4 units for every 3 units it moves to the right. This visual representation can help you quickly understand and graph a linear equation.
Plotting Points
To draw a line, you need to start by plotting points. Begin with the y-intercept because it is straightforward to find. The y-intercept \( b \) is the point where the line crosses the y-axis, which makes \( x = 0 \) by definition.

Taking the equation \( y = -\frac{4}{3}x + 3 \), we find the y-intercept at the point \( (0, 3) \). Next, use the slope to find more points. The slope \( -\frac{4}{3} \) tells us to move down 4 units and right 3 units from \( (0, 3) \). After moving as described, plot the second point at \( (3, -1) \).

These two points are enough to define the line, but you can plot additional points for accuracy.
Graphing Lines
After plotting two points, the next step is to graph the line. Use a ruler to connect the points \( (0, 3) \) and \( (3, -1) \) with a straight line. Make sure the line extends in both directions, as a line technically goes on forever.

Label the equation \( y = -\frac{4}{3}x + 3 \) on the graph to make it clear what line you have graphed. Also, mark your points clearly.

By only using the slope-intercept form and plotting 2 points, you are able to graph any linear equation. Practice these steps to become more comfortable with graphing lines on your own.

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Most popular questions from this chapter

Use a graphing calculator to find the function values. $$p(a)=\frac{2}{a}-a^{2}$$ a) \(p\left(\frac{1}{8}\right)\) b) \(p(-0.5)\)

Find the function values. $$g(x)=2 x+3$$ a) \(g(0)\) b) \(g(-4)\) c) \(g(-7)\) d) \(g(8)\) e) \(g(a+2)\) f) \(g(a)+2\)

Find the function values. $$h(x)=3 x-2$$ a) \(h(4)\) b) \(h(8)\) c) \(h(-3)\) d) \(h(-4)\) e) \(h(a-1)\) f) \(h(a)-1\)

Explain why the domain of the function given by \(f(x)=\frac{x+3}{2}\) is \(\mathbb{R},\) but the domain of the function given by \(g(x)=\frac{2}{x+3}\) is not \(\mathbb{R}\).

Write a slope-intercept equation of the line whose graph is described. Perpendicular to the graph of \(2 x+y=0\) \(y\) -intercept \((0,0)\)
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