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Chapter 2: Problem 122

For the following exercises, sketch the graph of each equation. $$h(x)=\frac{1}{3} x+2$$

Short Answer

Expert verified
The graph is a straight line with y-intercept 2 and slope \(\frac{1}{3}\).

Step by step solution

01

Identify the Equation Type

The given equation is in the slope-intercept form, which is \( y = mx + b \). Here, \( m = \frac{1}{3} \) (the slope) and \( b = 2 \) (the y-intercept). This means it represents a straight line.
02

Determine the Y-Intercept

The y-intercept \( (0, b) \) is where the graph intersects the y-axis. For the equation \( h(x) = \frac{1}{3}x + 2 \), the y-intercept is \( (0, 2) \). Plot this point on the y-axis.
03

Determine the Slope

The slope \( m = \frac{1}{3} \) means that for every 1 unit increase in \( x \), \( y \) increases by \( \frac{1}{3} \) units. From the y-intercept \( (0, 2) \), move right 1 unit (to \( x = 1 \)) and up \( \frac{1}{3} \) units to reach the next point, \( (1, 2.33) \). Plot this second point.
04

Sketch the Line

Draw a straight line through the points \( (0, 2) \) and \( (1, 2.33) \). This line is the graph of the equation \( h(x) = \frac{1}{3}x + 2 \). Make sure the line extends in both directions and through the plotted points.

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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 fundamental method for writing the equation of a line. It is expressed as \( y = mx + b \). The beauty of this form is in its simplicity and clarity. It provides two pieces of crucial information about the line at a glance:
  • The slope \( m \), which indicates how steep the line is.
  • The y-intercept \( b \), showing where the line crosses the y-axis.
By using the slope-intercept form, you can easily convert an equation into a graphed line on a coordinate plane. For example, given the equation \( h(x) = \frac{1}{3} x + 2 \), you immediately identify the slope as \( \frac{1}{3} \) and the y-intercept as \( 2 \). Thus, the form not only clarifies the representation of a line but also provides a definitive starting point for plotting the line graphically.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It is represented as \((0, b)\) in the slope-intercept form equation \( y = mx + b \). Here, \( b \) is the specific value of the y-intercept. This point is crucial because it anchors the line on the graph.
For the equation \( h(x) = \frac{1}{3}x + 2 \), the y-intercept is \( (0, 2) \). This means the line will pass through the point \( (0, 2) \) on the y-axis. To visualize this, plot the point directly on the y-axis at \( y = 2 \).
Understanding the y-intercept helps quickly draft the line without needing complex calculations, forming the foundation for accurately sketching the graph.
Slope
The slope of a line, symbolized as \( m \) in the equation \( y = mx + b \), represents the line's steepness and direction. It is expressed as a ratio \( \frac{rise}{run} \), indicating how much the line 'rises' or 'falls' for each unit it 'runs' along the x-axis.
For the line \( h(x) = \frac{1}{3}x + 2 \), the slope \( \frac{1}{3} \) tells us that:
  • For every step you move right along the x-axis, the line moves up by \( \frac{1}{3} \).
  • The slope is positive, meaning the line inclines upwards as it moves from left to right.
By confirming the slope, you can find additional points on the line by moving horizontally, then vertically according to this ratio. It ensures you maintain accuracy as you draw the graph beyond the y-intercept.
Coordinate Plane
The coordinate plane is a two-dimensional space, defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Together, these axes form a grid that allows for plotting points and graphing relationships between variables.
To graph the equation \( h(x) = \frac{1}{3}x + 2 \), start by marking the y-intercept \( (0, 2) \) on the y-axis. Using the slope \( \frac{1}{3} \), determine another point by moving from the y-intercept: one unit right to \( x = 1 \) and \( \frac{1}{3} \) unit up to \( y = 2.33 \).
  • Label your axes for clarity.
  • Use these points to draw a line, ensuring it extends through the plotted points.
  • The intersection between the axes at (0,0) is the origin, a reference point for all coordinates plotted.
Mastering the coordinate plane provides a versatile skill for representing linear equations visually, assisting you in conceptualizing their relationships and interactions.

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Most popular questions from this chapter

For the following exercises, consider the data in Table 2.18, which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year. $$\begin{array}{|c|c|c|c|c|c|}\hline \text { Year } & {2000} & {2002} & {2005} & {2007} & {2010} \\ \hline \text { Percent Graduates } & {6.5} & {7.0} & {7.4} & {8.2} & {9.0} \\ \hline\end{array}$$ Based on the set of data given in Table 2.20, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places. $$\begin{array}{|c|c|c|c|c|c|}\hline x & {10} & {12} & {15} & {18} & {20} \\\ \hline y & {36} & {34} & {30} & {28} & {22} \\ \hline\end{array}$$

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years: $$(4,500,2000) ;(4,700,2001) ;(5,200,2003) ;(5,800,2006)$$ Predict when the population will hit 20,000.

A city's population in the year 1960 was \(287,500 .\) In 1989 the population was \(275,900 .\) Compute the rate of growth of the population and make a statement about the population rate of change in people per year.

Find the area of a triangle by the \(y\) axis, the line \(f(x)=10-2 x,\) and the line perpendicular to \(f\) that passes through the origin.

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years: $$(2500,2000),(2650,2001),(3000,2003),(3500,2006),(4200,2010)$$ Use linear regression to determine a function \(y\), where the year depends on the population. Round to three decimal places of accuracy.
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