Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 16
Sketch the graph of the function. $$ f(t)=|t|+t $$
Short Answer
Expert verified
The graph is a line y=0 for t<0, and a line y=2t for t鈮�0, meeting at (0,0).
Step by step solution
01
Understand the Function
The function given is \( f(t) = |t| + t \). This is a piecewise function, combining an absolute value, \(|t|\), with a linear component, \(t\). To sketch it, we need to consider how these two components interact for different values of \(t\).
02
Break Down the Absolute Value
The absolute value \(|t|\) affects the function differently depending on \(t\). For \(t \geq 0\), \(|t| = t\), and for \(t < 0\), \(|t| = -t\). This means the function \(f(t)\) can be rewritten piecewise as: \[ f(t) = \begin{cases} 2t, & \text{if } t \geq 0 \ 0, & \text{if } t < 0 \end{cases}\]
03
Identify Key Points
Consider specific values to get crucial points of the graph. For instance, at \(t = 0\), \(f(0) = 0\). For \(t > 0\), the function simplifies to \(f(t) = 2t\), so \(f(1) = 2\), \(f(2) = 4\), etc. For \(t < 0\), the function simplifies to \(f(t) = 0\), so \(f(-1) = 0\), \(f(-2) = 0\).
04
Sketch the Graph
Plot the points identified: start from \((0,0)\), which is a turning point. For \(t \geq 0\), draw a line with a slope of 2 passing through \((0,0)\) upwards. For \(t < 0\), the function is constant and equal to zero, forming a horizontal line at \(y=0\).
05
Analyze the Graph
Observe that the graph is a line running along the x-axis for negative \(t\) and rises with a slope of +2 for non-negative \(t\). The function has a corner at \(t=0\), where the slope changes abruptly.
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.
Absolute Value Function
An absolute value function is a type of piecewise function that affects the input, or the "argument," by making sure its output value is non-negative. The absolute value of a number \(|t|\) represents its distance from zero on a number line, without considering its direction. It effectively "flips" any negative values into positive ones. For the function we are considering, \(|t|\), it works as follows:
- For \(t \geq 0\), \(|t| = t\), because positive numbers remain unchanged by the absolute value.
- For \(t < 0\), \(|t| = -t\), because negative numbers become positive when they are flipped by the absolute value operation.
Piecewise Function
Piecewise functions are functions that have different expressions based on the input value; they are defined in 'pieces' using multiple sub-functions. A great way to understand piecewise functions is to break them down according to the conditions given for the input variable. For example, in our function \(f(t) = |t| + t\), we derived it into a piecewise representation:
- For \(t \geq 0\), the function becomes \(f(t) = 2t\) because \(|t| + t = t + t = 2t\).
- For \(t < 0\), the function simplifies to \(f(t) = 0\) because \(|t| + t = (-t) + t = 0\).
Linear Function
A linear function is a function whose graph is a straight line. It has a general form of \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. In the simplified parts of our piecewise function, the segment \(f(t) = 2t\) is a linear function:
- The slope, \(m=2\), indicates that for each horizontal step (increase in \(t\)), the function's value goes up by \(2\) units vertically. This steepness of the line is visually observable.
- The y-intercept is \(b=0\), meaning the line crosses the origin \((0,0)\). This occurs because no constant term is added in the equation \(2t\).
