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Chapter 2: Problem 61

Graph the linear function \(f\) on a domain of \([-0.1,0.1]\) for the function whose slope is 75 and \(y\) -intercept is \(-22.5 .\) Label the points for the input values of \(-0.1\) and \(0.1 .\)

Short Answer

Expert verified
Plot the linear function \( f(x) = 75x - 22.5 \), intersecting the y-axis at -22.5 and passing through points (-0.1, -30) and (0.1, -15).

Step by step solution

01

Understand the Components of the Linear Function

The function is given in the slope-intercept form, which is generally expressed as \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept of the function. Here, the slope \( m = 75 \) and the y-intercept \( b = -22.5 \).
02

Write the Equation of the Function

Using the known slope and y-intercept, we can write the equation of our linear function as \( f(x) = 75x - 22.5 \).
03

Calculate Values for Boundary Points

We need to calculate the function values at the boundaries of the domain, \(-0.1\) and \(0.1\).- For \( x = -0.1 \): \[ f(-0.1) = 75(-0.1) - 22.5 = -7.5 - 22.5 = -30 \]- For \( x = 0.1 \): \[ f(0.1) = 75(0.1) - 22.5 = 7.5 - 22.5 = -15 \]
04

Plot the Function on the Given Domain

On a coordinate plane, plot the points calculated:- \( (-0.1, -30) \)- \( (0.1, -15) \)Draw a straight line through these points to represent the function \( f(x) = 75x - 22.5 \).
05

Label the Points on the Graph

Ensure the graph includes labels at the points you calculated:- Point A: \( (-0.1, -30) \)- Point B: \( (0.1, -15) \)These labels help indicate the calculated values at the boundaries of the domain.

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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 of a linear function is a straightforward way of expressing linear equations. It is written as \( f(x) = mx + b \). This form allows you to easily identify key characteristics of a line. Here, \( m \) represents the slope, and \( b \) designates the y-intercept.
  • The slope \( m \) reveals how steep the line is. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
  • The y-intercept \( b \) tells us where the line crosses the y-axis. This is the point where \( x = 0 \).
Using the equation, you can quickly visualize the line and how it behaves in relation to the axes.
Domain
The domain of a function refers to all the possible input values \( x \) for which the function is defined. In our exercise, we were given a specific domain: \([-0.1, 0.1]\). This means we only consider inputs from \(-0.1\) to \(0.1\) for our linear function.
  • The domain is important because it restricts the portions of the function we look at or graph.
  • In real-world applications, domains can be restricted due to practical reasons, such as time intervals or physical boundaries.
By calculating function values only within the given domain, we focus on a specific segment of the line.
Graphing Linear Equations
Graphing a linear equation involves plotting points and connecting them to form a straight line. Here's how to do it for the function in our exercise:
1. Calculate specific values of \( f(x) \) at both ends of the domain.2. In our case, for \( x = -0.1 \) and \( x = 0.1 \), the corresponding points were \((-0.1, -30)\) and \((0.1, -15)\).3. Plot these points on the graph.4. Draw a straight line through these points.
  • Graphing linear equations visually represents how the value of \( f(x) \) changes with \( x \).
  • This method is powerful because it gives a clear image of the relationship between the variables.
Slope
The slope of a line, represented as \( m \) in the slope-intercept form, indicates the direction and steepness of the line. In our exercise, the slope was \( 75 \), which is quite steep.
  • A larger slope value means the line is steeper.
  • A positive slope, like \( 75 \), shows the line rises as it moves from left to right.
  • If the slope were negative, the line would fall instead.
Slope is a key aspect because it directly affects the angle of the line and thus how quickly the function value changes as \( x \) increases or decreases.
Y-intercept
The y-intercept is the point where a line crosses the y-axis, and is indicated by \( b \) in the equation \( f(x) = mx + b \). For the given linear function, the y-intercept was \( -22.5 \).
  • This means that when \( x = 0 \), \( f(x) = -22.5 \).
  • The y-intercept gives us a starting point, guiding where the graph begins along the y-axis.
Understanding the y-intercept helps to better position the graph correctly on the coordinate plane.

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