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Chapter 6: Problem 17

A linear function is given. Evaluate the function at the indicated values and then graph the function over its given domain. $$F(x)=2+3.5 x, x=-2,-1,0,1,2 ; F(0), F(1)$$

Short Answer

Expert verified
\(F(0) = 2\), \(F(1) = 5.5\). Graph by plotting and connecting \((-2, -5), (-1, -1.5), (0, 2), (1, 5.5), (2, 9)\).

Step by step solution

01

Understanding the Function

The given linear function is expressed as \(F(x) = 2 + 3.5x\). It is in the form of \(F(x) = mx + b\), where \(m = 3.5\) (the slope) and \(b = 2\) (the y-intercept).
02

Evaluating the Function at Specific Points

To evaluate \(F(x)\) at the given points, substitute each value into the function:- \(F(-2) = 2 + 3.5(-2) = 2 - 7 = -5\)- \(F(-1) = 2 + 3.5(-1) = 2 - 3.5 = -1.5\)- \(F(0) = 2 + 3.5(0) = 2\)- \(F(1) = 2 + 3.5(1) = 2 + 3.5 = 5.5\)- \(F(2) = 2 + 3.5(2) = 2 + 7 = 9\)
03

Graphing the Function

To graph \(F(x)\), plot the points calculated in Step 2: \((-2, -5), (-1, -1.5), (0, 2), (1, 5.5), (2, 9)\). Connect these points with a straight line since this is a linear function.
04

Final Evaluation

From Step 2, we find that \(F(0) = 2\) and \(F(1) = 5.5\). These values are important as they may be specifically asked for in further analysis.

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Key Concepts

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

Function Evaluation
To evaluate a linear function, we start with understanding its form, which in this exercise is given as \( F(x) = 2 + 3.5x \). This function follows the standard linear equation format \( F(x) = mx + b \). In this formula, \( m \) is the slope and \( b \) is the y-intercept. Function evaluation is a process where we plug in specific values of \( x \) to find corresponding values of \( F(x) \). This helps us understand the behavior of the function at different points.

Here's how we conduct function evaluation for this exercise:
  • Substitute \( x = -2 \), and calculate: \( F(-2) = 2 + 3.5(-2) = -5 \)
  • Substitute \( x = -1 \), and calculate: \( F(-1) = 2 + 3.5(-1) = -1.5 \)
  • For \( x = 0 \), find: \( F(0) = 2 + 3.5(0) = 2 \)
  • Substitute \( x = 1 \), resulting in: \( F(1) = 5.5 \)
  • Finally, for \( x = 2 \), evaluate: \( F(2) = 9 \)
This step-by-step substitution shows how the function changes with different values of \( x \), providing a numerical view of the function's linear nature.
Graphing Functions
Graphing a linear function provides a visual representation of the function's behavior. For the function \( F(x) = 2 + 3.5x \), once we've evaluated it at several points, we can use these results to plot a graph.

Let's break down the graphing process:
  • Identify the points you want to plot, which should be the same evaluated points: \((-2, -5), (-1, -1.5), (0, 2), (1, 5.5), (2, 9)\).
  • Plot these points on a coordinate plane.
  • Draw a straight line through these points. Since the function is linear, its graph will always be a straight line.
Graphing helps in visualizing the increase or decrease of function values, making it easier to understand trends and relationships in data. The straight line indicates a constant rate of change, due to the linear nature of the function.
Slope and Y-Intercept
Understanding the slope and y-intercept of a linear function is crucial, as it provides insights into the function's behavior and direction.
  • The **slope** \( m \) determines the steepness and direction of the line. For the function \( F(x) = 2 + 3.5x \), the slope \( m = 3.5 \) indicates that for every unit increase in \( x \), the value of \( F(x) \) increases by 3.5 units. A positive slope means the line rises as \( x \) increases.
  • The **y-intercept** \( b \) is the point where the line crosses the y-axis, which is at \( (0, 2) \) for this function. This is the value of \( F(x) \) when \( x = 0 \), showing where the function starts on the y-axis.

These concepts help us interpret the graph:
  • A steeper slope indicates a more rapid increase or decrease in function value.
  • The y-intercept provides a starting point, helping to anchor the graph on the y-axis.
Understanding these parameters allows us to quickly sketch the function and anticipate its behavior without detailed calculations.

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Most popular questions from this chapter

Make a table of function values using the given discrete domain values. Write the values as ordered pairs and then graph the function. $$f(x)=x, \quad x=-2,-1,0,1,2$$

Consider the following scenario: From 1990 to 1999 , the amount spent to purchase a car exported to the United States increased at rate of \(\$ 1780\) each successive year. The increase in amount spent to purchase a car exported to the United States from 1990 to 1999 can be represented by the rate function $$f(x)=\left\\{\begin{array}{cl} 1.78, & 0 \leq x \leq 9 \\ 0, & \text { otherwise } \end{array}\right.$$ where \(x\) represents the number of years since 1990 and \(f(x)\) represents the rate of change in amount in thousands of dollars per year. Calculate the area under \(f(x)\) on the interval \(5 \leq x \leq 8\) and interpret the result.

Solve the following applications involving the area of a rectangle. A popular smartphone has a screen height of 3.5 inches and a width of 1.9 inches. Calculate the screen's area.

Find the area under \(f(x)\) on the indicated interval. Round the area to two decimal places as necessary. $$f(x)=\left\\{\begin{array}{cl} 1.1 x, & 0 \leq x \leq 11 \\ 0, & \text { otherwise } \end{array} \text { on the interval } 5 \leq x \leq 10\right.$$

use the following information: James's wireless phone plan has a fee of \(\$ 15\) per month for the first 500 text messages and a \(\$ 0.02\) charge for each text thereafter. The monthly cost of text messaging is given by the piece wise defined function $$f(x)=\left\\{\begin{aligned} 15, & x=0,1,2, \ldots, 500 \\\ 5+0.02 x, & x=501,502, \ldots \end{aligned}\right.$$ where \(x\) represents the number of text messages and \(f(x)\) is cost in dollars. Evaluate \(f(550)\) and interpret the result.
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