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Chapter 2: Problem 273

Sketch a graph of the linear function \(f(t)=2 t-5\).

Short Answer

Expert verified
Graph the line passing through (0, -5) with a slope of 2.

Step by step solution

01

Identify the Slope and Y-Intercept

The given linear function is in the form \( f(t) = mt + b \), where \( m \) is the slope and \( b \) is the y-intercept. For the function \( f(t) = 2t - 5 \), the slope \( m \) is 2 and the y-intercept \( b \) is -5. This means that the line will cross the y-axis at the point (0, -5) and will rise 2 units for every 1 unit of run to the right.
02

Plot the Y-Intercept

Start the graph by plotting the y-intercept point on the coordinate plane. From Step 1, the y-intercept is at the point (0, -5). Locate this point on the graph, which is on the vertical y-axis.
03

Use the Slope to Find Another Point

Using the slope of the line, which is 2, we can find another point on the line. The slope \( m = \frac{2}{1} \) means from the y-intercept (0, -5), move 1 unit to the right (positive direction along the x-axis) and then 2 units up (positive direction along the y-axis). This brings us to the point (1, -3). Plot this point.
04

Draw the Line

With the two points (0, -5) and (1, -3) plotted on the graph, use a ruler to draw a straight line through these points. Extend the line beyond these points and in both directions to better visualize the linear function.

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Key Concepts

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

Slope
The slope of a linear function is a crucial concept that helps us understand how "steep" a line is and in which direction it travels on the coordinate plane. In the linear function, the slope is represented by the letter \( m \). It tells us the change in the vertical direction (y-axis) for each unit change in the horizontal direction (x-axis).
For the function \( f(t) = 2t - 5 \), the slope \( m \) is 2. This means that for every 1 unit you move to the right along the x-axis, you move 2 units up along the y-axis.
Remember:
  • If the slope is positive, like in our example, the line ascends as it moves from left to right.
  • A negative slope would mean the line descends, moving downward from left to right.
  • A slope of zero would result in a perfectly horizontal line.
Understanding the slope helps you quickly sketch the line on a graph once you have the starting point, or the y-intercept.
Y-Intercept
The y-intercept is a key part of understanding where a line intersects the y-axis on a graph. It's represented in the equation \( f(t) = mt + b \) by the term \( b \). This value tells us precisely where the line will "hit" the y-axis.
For the function \( f(t) = 2t - 5 \), the y-intercept \( b \) is -5. This tells us that when \( t \) is zero, the value of the function is -5. So, the line crosses the y-axis at the point (0, -5).
Here鈥檚 what鈥檚 helpful to remember:
  • The y-intercept provides a starting point for your line on the graph.
  • It's the initial value of the function when \( t \) (or x in a different context) is zero.
  • To find it graphically, look for where the line cuts across the y-axis.
Knowing the y-intercept is the first step in drawing your linear function on a coordinate plane.
Coordinate Plane
The coordinate plane is the grid system that helps plot points and visualize functions. It consists of two perpendicular lines, known as axes:
  • The horizontal axis, or x-axis, typically represents independent variable values.
  • The vertical axis, or y-axis, represents the dependent variable values based on inputs from the x-axis.
In our example, plotting the linear function \( f(t) = 2t - 5 \) involves placing points on this plane. By understanding the coordinates,
  • we start at the y-intercept (0, -5),
  • use the slope to determine further points, such as moving to (1, -3).
The coordinate plane is crucial because it gives a visual way of understanding mathematical relationships. It helps connect what's often abstract into something we can see and draw.
Graphing
Graphing a linear function involves drawing its representation on the coordinate plane. This process helps transform numerical relationships into visual data. For the function \( f(t) = 2t - 5 \), here鈥檚 a breakdown:1. Plot the y-intercept (0, -5) on the y-axis.2. Use the slope for direction: from (0, -5), move 1 unit right and 2 units up to find the next point (1, -3).3. With two points plotted, draw a straight line through them. Extend this line in both directions.
Benefits of graphing:
  • Makes it easier to see the trend and direction of the function.
  • Allows visual checking of points along the line.
  • Facilitates understanding of intercepts and slopes visually.
Graphing turns numbers and equations into pictures, bridging numerical and visual learning.

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Most popular questions from this chapter

Draw a scatter plot for the data in Table 2.15. Then determine whether the data appears to be linearly related. $$\begin{array}{|c|c|c|c|c|c|}\hline 0 & {2} & {4} & {6} & {8} & {10} \\\ \hline-105 & {-50} & {1} & {55} & {105} & {160} \\ \hline \end{array}$$

Determine whether the following function is increasing or decreasing. \(f(x)=-2 x+5\)

Graph the linear function \(f\) on a domain of \([-10,10]\) for the function whose slope is \(\frac{1}{8}\) and \(y\) -intercept is \(\frac{31}{16} .\) Label the points for the input values of \(-10\) and \(10 .\)

The number of people afflicted with the common cold in the winter months dropped steadily by 25 each year since 2002 until 2012. In 2002, 8,040 people were inflicted. Find the linear function that models the number of people afflicted with the common cold \(C\) as a function of the year, \(t\). When will less than 6,000 people be afflicted?

A clothing business finds there is a linear relationship between the number of shirts, \(n,\) it can sell and the price, \(p,\) it can charge per shirt. In particular, historical data shows that \(1,000\) shirts can be sold at a price of \(\$ 30\) , while \(3,000\) s shirts can be sold at a price of \(\$ 22 .\) Find a linear equation in the form \(p(n)=m n+b\) that gives the price \(p\) they can charge for \(n\) shirts.
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