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Chapter 1: Problem 26

Graph each function "by hand." [Note: Even if you have a graphing calculator, it is important to be able to sketch simple curves by finding a few important points.] $$ f(x)=-3 x+5 $$

Short Answer

Expert verified
To graph \(f(x) = -3x + 5\), plot the y-intercept \((0, 5)\) and use the slope \(-3\) to plot \((1, 2)\). Connect the points to form the line.

Step by step solution

01

Identify the Type of Function

The given function \(f(x) = -3x + 5\) is a linear function. Linear functions produce straight lines when graphed. This function is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Determine the Slope and Y-intercept

For the function \(f(x) = -3x + 5\), the slope \(m\) is \(-3\) and the y-intercept \(b\) is \(5\). This means the line will cross the y-axis at \(5\), and for each unit increase in \(x\), \(y\) will decrease by 3 units.
03

Plot the Y-intercept

Begin the graph by plotting the y-intercept. On a coordinate plane, place a point at \( (0, 5)\), since this is where the graph intersects the y-axis.
04

Use the Slope to Plot a Second Point

Starting from the y-intercept \((0, 5)\), use the slope \(-3\) to determine another point. From \((0, 5)\), move 1 unit to the right (positive x-direction) and 3 units down (negative y-direction) to plot the point \((1, 2)\).
05

Draw the Line

With the points \((0, 5)\) and \((1, 2)\) plotted, draw a straight line through these points. This line represents the graph of the function \(f(x) = -3x + 5\).
06

Verify the Line

Double-check another point by using the function to ensure accuracy. For example, calculate \(f(2) = -3(2) + 5 = -6 + 5 = -1\). Plot this point at \((2, -1)\), and confirm it lies on the line.

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Key Concepts

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

Linear Functions
Linear functions are a key concept in algebra and mathematics. They are equations that form straight lines when graphed on a coordinate plane. The simplest example of a linear function is of the form \( f(x) = mx + b \). This formula represents a straight line where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept, which is where the line crosses the y-axis.
Linear functions have only one variable raised to the first power, and they do not include curves, parabolas, or any other non-linear shapes. Understanding these fundamental characteristics of linear functions is essential for interpreting and graphing them effectively.
Slope-Intercept Form
The slope-intercept form is a way of writing a linear equation so it's easy to graph. It's represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the intercept. Let's break this down:
  • The slope \( m \) tells us how steep the line is. A positive slope means the line rises, while a negative slope means it falls.
  • The y-intercept \( b \) gives the starting point of the graph on the y-axis.
For example, in the function \( f(x) = -3x + 5 \), the slope \(-3\) means the line will drop 3 units vertically for every 1 unit it moves horizontally. The y-intercept \(5\) means the line will cross the y-axis at the point (0,5). This form makes it simple to graph the function by just plotting the y-intercept and using the slope to find another point.
Graphing Techniques
Graphing techniques for linear functions involve plotting points and using the line properties. Here鈥檚 a simple way to approach it:
  • Start by identifying the y-intercept \( b \). For \( f(x) = -3x + 5 \), place a point at the y-intercept (0,5).

  • Next, apply the slope \( m \). With a slope of \(-3\), you move one unit right and then three units downward to mark another point such as (1,2).

  • Draw a line passing through both points, ensuring it extends indefinitely in both directions.

  • Verify accuracy by checking additional points. For example, substitute \( x = 2 \) to confirm it falls on the line.

This systematic approach makes graphing straightforward, translating the algebraic representation into a visual one seamlessly.

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Most popular questions from this chapter

How will the graph of \(y=(x+3)^{3}+6\) differ from the graph of \(y=x^{3}\) ? Check by graphing both functions together.

How do the graphs of \(f(x)\) and \(f(x+10)+10\) differ?

True or False: If \(f(x)=m x+b\), then \(f(x+h)=f(x)+m h .\)

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=x^{3}+x ; g(x)=\frac{x^{4}+1}{x^{4}-1} $$

Find the slope (if it is defined) of the line determined by each pair of points. \((0,-1)\) and \((4,-1)\)
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