Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 29
Sketch the graph of \(f(t)=|5-2 t|\) (absolute value) for |t \(\mid \leqslant 2\) and find its slopes and range.
Short Answer
Expert verified
For \(|t| \leq 2\), the slope is \(-2\) and range is \([1, 9]\).
Step by step solution
01
Identify the Expression inside the Absolute Value
The function given is \( f(t) = |5 - 2t| \). Inside the absolute value, we have the linear expression \( 5 - 2t \). The absolute value affects how we analyze changes in sign of this expression.
02
Determine When the Expression Changes Sign
Set the expression inside the absolute value to zero: \( 5 - 2t = 0 \). Solving for \( t \), we get \( t = \frac{5}{2} = 2.5 \). This is where the expression changes from positive to negative or vice versa.
03
Analyze the Piecewise Function
Because of the absolute value, \( f(t) \) can be written as a piecewise function:- For \( t < 2.5 \), \( f(t) = 5 - 2t \) - For \( t > 2.5 \), \( f(t) = -(5 - 2t) = 2t - 5 \) This transformation helps in sketching the two linear segments.
04
Sketch the Graph
On a graph, plot:- The line \( f(t) = 5 - 2t \) for \( t \leq 2.5 \)- The line \( f(t) = 2t - 5 \) for \( t \geq 2.5 \)Ensure continuity at \( t = 2.5 \), where both expressions yield \( 0 \). For \( |t| \leq 2 \), the function \( f(t) \) remains as \( 5 - 2t \).
05
Find the Slope of Each Segment
The slope of \( f(t) = 5 - 2t \) is \(-2\) for \( t \leq 2.5 \) and we are using this slope for \( |t| \leq 2 \). Since \( t > 2 \) is not applicable here due to the condition on \( t \), the slope beyond \( 2 \) for the graph is not considered.
06
Determine the Range
For \( |t| \leq 2 \), calculate \( f(t) \):- At \( t = -2 \), \( f(-2) = |5 + 4| = 9 \)- At \( t = 2 \), \( f(2) = |5 - 4| = 1 \)Thus, the range of \( f(t) \) for \( |t| \leq 2 \) is \([1, 9]\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Piecewise Functions
In mathematics, a piecewise function is a function defined by multiple sub-functions, each of which applies to a particular interval of the main function's domain. Piecewise functions let us express complex functions in a simpler manner by breaking them into parts. Consider the function given: \[ f(t) = |5 - 2t| \]The absolute value affects how we handle this function. We convert it into a piecewise function, which depends on the sign of the expression inside the absolute value:
- For \( t < 2.5 \), \( f(t) = 5 - 2t \), because the expression is positive.
- For \( t > 2.5 \), \( f(t) = 2t - 5 \), because the expression inside the absolute value is negative, hence the absolute value takes the negative of \(5 - 2t\).
Graph Sketching
Graph sketching is a technique used to visually represent mathematical functions, focusing on important features such as intercepts, slopes, and transitions. To sketch the graph of a piecewise function like \[ f(t) = |5 - 2t| \]we first need to plot the linear segments described in the piecewise function. For \(-2 \leq t \leq 2\), we only consider the segment \[f(t) = 5 - 2t\].Here's how you can graphically interpret the function:
- Plot the line \( f(t) = 5 - 2t \) from \( t = -2 \) to \( t = 2 \), forming a linear segment.
- Ensure continuity at each transition. Since \( t = 2.5 \) is outside our range of \(|t| \leq 2\), we only focus up to \( t = 2 \).
Function Range
The range of a function is the set of all possible output values it can produce. To determine the range of \[ f(t) = |5 - 2t| \]for \(|t| \leq 2\), we calculate the function at the boundaries of its domain:
- When \(t = -2\), the function value is \(f(-2) = |5 + 4| = 9\).
- When \(t = 2\), the function value is \(f(2) = |5 - 4| = 1\).
Slope Analysis
Slope analysis helps us determine the steepness of a line and its direction on the graph. For the function: \[ f(t) = 5 - 2t \]we analyze the slope within the interval \(-2 \leq t \leq 2\). The slope is calculated as follows:- The formula for the line is \(f(t) = 5 - 2t\), which makes the slope -2 by the coefficient of \(t\).This slope indicates that the line is decreasing, with the rate of change indicating how quickly the function value decreases as \(t\) increases.
- Between the range \(-2 \leq t \leq 2\), the function maintains this slope.
