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Chapter 8: Problem 83

Graph each function. See Objective 5. $$ f(x)=2 x-1 $$

Short Answer

Expert verified
Graph the line passing through (0, -1) and (1, 1) with slope 2.

Step by step solution

01

Identify the slope and y-intercept

The function is in the slope-intercept form, which is given by \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = 2 \) and \( b = -1 \). This means the line has a slope of 2 and a y-intercept at (0, -1).
02

Plot the y-intercept

Start by plotting the y-intercept on the graph. Since \( b = -1 \), place a point on the y-axis at (0, -1).
03

Use the slope to plot another point

The slope \( m = 2 \) can be interpreted as 'rise over run,' which is \( \frac{2}{1} \). From the y-intercept point (0, -1), move up 2 units and 1 unit to the right. This gives the second point, (1, 1).
04

Draw the line

With the points (0, -1) and (1, 1) plotted, draw a straight line through these points extending in both directions. This represents the graph of the function \( f(x) = 2x - 1 \).

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Key Concepts

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

Understanding the Slope-Intercept Form
Graphing a linear function can be straightforward if we start with the slope-intercept form. This is a widely used format of a linear equation expressed as \( f(x) = mx + b \). Here, \( m \) and \( b \) play crucial roles in shaping the line on a graph.
The term \( m \) corresponds to the slope of the line. The slope indicates how steep the line is and the direction it tilts. Meanwhile, \( b \) is the y-intercept, which shows where the line crosses the y-axis.
  • **Key Components:** \( m \) (slope) and \( b \) (y-intercept)
  • **Format Example:** \( f(x) = 2x - 1 \)
Understanding this form allows you to quickly identify these crucial components automatically upon seeing a linear equation.
Grasping the Slope
The slope of a line is a measure of its steepness or incline, often described as 'rise over run'. In simple terms, it tells us how much the line goes up (or down) for every step it takes horizontally.
Let's take the example \( y = 2x - 1 \), where the slope \( m \) is 2. This means for every unit you move to the right, the line goes up 2 units.
  • **Slope as a Fraction:** \( m = \frac{2}{1} \)
  • **Interpretation:** Move up 2, and across 1 to the right
Understanding slope helps you plot additional points along the line by simply following the 'rise over run' rule. It is essential in accurately graphing lines and describing their positions on a plane.
Navigating the Y-intercept
The y-intercept is a key feature of the slope-intercept form and represents the point where a line crosses the y-axis. This means that at some point, the x-coordinate is zero, making it easy to plot this point as a starting reference for graphing.
In the function \( f(x) = 2x - 1 \), the y-intercept \( b \) is \(-1\). So, its graph crosses the y-axis at the point (0, -1).
  • **Location:** Point (0, -1) on the graph
  • **Importance in Graphing:** Provides a starting point to draw the line
Knowing the y-intercept gives you the initial anchor point on the graph, making it the foundation of plotting linear equations. Together with the slope, it defines the unique position and path of the line.

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Most popular questions from this chapter

Decongestants. The temperature in degrees Celsius that is equivalent to a temperature in degrees Fahrenheit is given by the linear function \(C(F)=\frac{5}{9}(F-32) .\) Refer to the label from a bottle of decongestant shown below. Use this function to find the low and high temperature extremes, in degrees Celsius, in which the bottle should be stored. DIRECTIONS: Adults and children 12 years of age and over: Two teaspoons every 4 hours. DO NOT EXCEED 6 DOSES IN A 24 -HOUR PERIOD. Store at a controlled room temperature between \(68^{\circ} \mathrm{F}\) and \(77^{\circ} \mathrm{F}\).

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{t}{t^{2}+5 t+6}-\frac{2}{t^{2}+3 t+2}$$

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{a x+a y+b x+b y}{x^{3}-27} \cdot \frac{x^{2}+3 x+9}{x c+x d+y c+y d}$$

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{18 m^{4}}{36 m^{4}-9 m^{3}}$$

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope \(\frac{1}{4},\) passes through \((8,1)\)
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