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Chapter 0: Problem 29

Graph each function. $$ f(x)=2 x-5 $$

Short Answer

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

Step by step solution

01

Understand the Function

The function given is a linear function in the form of \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For \(f(x) = 2x - 5\), the slope \(m\) is 2 and the y-intercept \(b\) is -5.
02

Plot the Y-Intercept

Start by plotting the y-intercept on the graph. This is the point where the line crosses the y-axis. For \(f(x) = 2x - 5\), the y-intercept is -5, so plot the point (0, -5) on the graph.
03

Use the Slope to Find Another Point

The slope of the line tells us how to move from one point to another on the graph. A slope of 2 means for every 1 unit you move to the right (positive x-direction), you move 2 units up (positive y-direction). Starting from (0, -5), move 1 unit to the right to (1, -5), then 2 units up to (1, -3) and plot this point.
04

Draw the Line

Once you have two points, (0, -5) and (1, -3), draw a straight line through these points extending in both directions. This represents the graph of the function \(f(x) = 2x - 5\).
05

Verify Additional Points (Optional)

To ensure accuracy, you can substitute another x-value into the function to find a third point. For example, if \(x = 2\), then \(f(2) = 2(2) - 5 = -1\). Plot the point (2, -1) and check if it lies on your line.

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Key Concepts

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

Slope-Intercept Form
Linear functions often use a specific format known as the slope-intercept form. This is expressed as \(y = mx + b\), where:
  • \(m\) represents the slope of the line.
  • \(b\) is the y-intercept, indicating where the line crosses the y-axis.
In our example function \(f(x) = 2x - 5\), the slope \(m\) is 2 and the y-intercept \(b\) is -5. Understanding this formulation helps us not only in identifying key characteristics of a line but also in plotting it on a graph. The slope-intercept form makes it straightforward to grasp the behavior of the line rapidly鈥攂y showing how steep the line is and where it starts on the y-axis.
Y-Intercept
The y-intercept is a crucial point in the linear function, denoted as \(b\) in the slope-intercept equation. This is the point where the line touches or crosses the y-axis, meaning it occurs where \(x = 0\). For \(f(x) = 2x - 5\), the y-intercept is -5.
  • This results in the coordinate point \((0, -5)\) on the graph.
Starting with the y-intercept provides a rooted origin for sketching the rest of the line. It's the foundation from which the line grows based on its slope. Thus, plotting the y-intercept establishes the first coordinate essential for drawing the full line representation of the function.
Plotting Points
Plotting points on a graph are steps that bring mathematical equations to life through visuals. For our function, once you have the y-intercept at \((0, -5)\), use the slope to identify and plot additional points.
  • The slope of 2 indicates that for every unit moved to the right, you go up 2 units. Hence, from \((0, -5)\), moving right 1 unit gets you to \((1, -3)\).
  • Plotting a new point involves checking by substituting different x-values into the function.
Adding multiple strategic points, such as \((2, -1)\), allows you to verify the accuracy of your line. Connecting these points accurately draws the line representing the function \(f(x) = 2x - 5\) and brings a clearer understanding of its characteristics.

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