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Chapter 1: Problem 18

For each equation, find the slope \(m\) and \(y\) intercept \((0, b)\) (when they exist) and draw the graph. \(y=-\frac{1}{3} x+2\)

Short Answer

Expert verified
The slope is \(-\frac{1}{3}\) and the y-intercept is \(2\).

Step by step solution

01

Identifying the Equation Form

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

Determining the Slope

From the equation \(y = -\frac{1}{3}x + 2\), we identify \(m = -\frac{1}{3}\). This means that for every unit increase in \(x\), \(y\) decreases by \(\frac{1}{3}\).
03

Finding the Y-Intercept

In the equation \(y = -\frac{1}{3}x + 2\), the y-intercept \(b\) is equal to 2. Thus, the line crosses the y-axis at the point \((0, 2)\).
04

Plotting the Y-Intercept

Start by plotting the y-intercept on the graph, which is at the point \((0, 2)\).
05

Using the Slope to Plot Another Point

From the y-intercept \((0, 2)\), use the slope \(-\frac{1}{3}\) to find another point on the line by going down 1 unit and right 3 units, which lands you at the point \((3, 1)\).
06

Drawing the Line

Connect the points \((0, 2)\) and \((3, 1)\) with a straight line. This is the graph of the equation \(y = -\frac{1}{3}x + 2\).

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Key Concepts

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

Graphing linear equations
Graphing linear equations involves translating the math equation into a visual format on a coordinate plane. A linear equation in two variables, like the one we have here, forms a straight line when graphed. The basic form often used is the slope-intercept form, like this equation:
  • The equation is given as \( y = mx + b \), where:
    • \( m \) represents the slope of the line,
    • \( b \) is the y-intercept, or the point where the line crosses the y-axis.
Understanding this form makes it easy to graph since you immediately know where to start (the y-intercept) and how to plot another point using the slope. Once a couple of points are found using these parameters, you can accurately draw a line through them, creating a visual representation of the equation.
Finding the slope
Finding the slope of a line in a linear equation is a key step. In the slope-intercept form, \( y = mx + b \), the slope \( m \) provides vital information on how steep the line is and its direction.
  • If \( m \) is:
    • Positive, the line slants upwards from left to right.
    • Negative, the line slants downwards from left to right.
    • Zero, the line is horizontal.
  • The slope is a ratio indicating vertical change over horizontal change between any two points on the line.
    • This means that for each unit you move horizontally, you move the amount given by the slope vertically.
In our example, \( m = -\frac{1}{3} \), indicating that for every increase of 3 units in \( x \), \( y \) decreases by 1 unit. Thus, the line will have a gentle downward slope.
Y-intercept calculation
The y-intercept calculation is straightforward once you have your equation in slope-intercept form. The y-intercept is essentially the point where the line crosses the y-axis. This occurs when x is equal to zero.
  • In the equation \( y = mx + b \):
    • The y-intercept \( b \) is clearly visible as it is the number added or subtracted from the slope term.
  • For our example, the y-intercept is straightforward to find:
    • It is \( b = 2 \), meaning the line crosses the y-axis at the point \( (0, 2) \).
  • Knowing the y-intercept gives a firm starting point for graphing the equation, ensuring you place the line accurately on the graph.
Remember, after determining the y-intercept, you can use the slope to find other points along the line, helping to ensure accuracy in your graph.

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Most popular questions from this chapter

The following problems extend and augment the material presented in the text. For any \(x\), the function \(\operatorname{INT}(x)\) is defined as the greatest integer less than or equal to \(x\). For example, \(\operatorname{INT}(3.7)=3\) and \(\operatorname{INT}(-4.2)=-5\) a. Use a graphing calculator to graph the function \(y_{1}=\operatorname{INT}(x)\). (You may need to graph it in DOT mode to eliminate false connecting lines.) b. From your graph, what are the domain and range of this function?

Find the slope (if it is defined) of the line determined by each pair of points. \((2,-1)\) and \((2,5)\)

For each function, find and simplify \(f(x+h)\). $$ f(x)=3 x^{2}-5 x+2 $$

Use the TABLE feature of your graphing calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for values of \(x\) such as \(100,10,000,1,000,000\), and higher values. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter 4 .

For each pair of functions \(f(x)\) and \(g(x)\), find and fully simplify a. \(f(g(x))\) and b. \(g(f(x))\) $$ f(x)=x^{3}+1 ; \quad g(x)=\sqrt[3]{x-1} $$
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