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Chapter 9: Problem 5

Graph each of the following linear and quadratic functions. $$f(x)=-3 x$$

Short Answer

Expert verified
Graph the line using slope \(-3\); it passes through \((0,0)\) and \((1,-3)\).

Step by step solution

01

Identify the Function Type

First, we need to identify the type of function we are dealing with. The given function is \( f(x) = -3x \). This is a linear function because it has the form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.
02

Determine the Slope and Y-intercept

In the function \( f(x) = -3x \), the slope \( m \) is \(-3\) and the y-intercept \( b \) is \(0\). The line will pass through the origin, meaning the point \((0,0)\). The slope tells us the line rises or falls as we move along the x-axis.
03

Plot the Y-intercept

The y-intercept is where the line crosses the y-axis. Since the y-intercept is 0, plot the point \((0,0)\) on the graph.
04

Use the Slope to Find Another Point

The slope \(-3\) means that for each unit of increase in \( x \), the value of \( y \) will decrease by 3 units. From the y-intercept \((0,0)\), move right 1 unit to \( x = 1 \) and down 3 units to \( y = -3 \), plotting the point \((1, -3)\).
05

Draw the Line

With the points \((0,0)\) and \((1,-3)\) plotted, draw a straight line through these points. This represents the graph of the function \( f(x) = -3x \). Linearity means the line continues infinitely in both directions at the same slope.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a fundamental way to express linear equations. It is usually written as \( y = mx + b \). Here, \( m \) stands for the slope of the line, and \( b \) is the y-intercept. The y-intercept is where the line crosses the y-axis. In this form, understanding each component is crucial for graphing a line accurately.
  • The **slope** \( m \) indicates how steep the line is and in which direction it tilts.
  • The **y-intercept** \( b \) shows the starting point of the line when \( x = 0 \).
By using the slope-intercept form, you can quickly pinpoint two vital characteristics of a line: its angle of inclination and where it intersects the y-axis. This makes it an excellent tool for both graphing and understanding linear relationships.
Graphing Techniques
Graphing linear functions such as \( f(x) = -3x \) involves a step-by-step approach to plotting points and connecting them correctly. Start by identifying key features like the slope and y-intercept, which guide you in plotting the points on a graph.
  • **Plot the y-intercept**: This is your starting point. In our function, it's at \((0,0)\).
  • **Use the slope**: The slope of \(-3\) indicates that for each step right on the x-axis, you move down three units on the y-axis.
Once you plot the starting point, apply the slope to find your next point. Draw a straight line through these plotted points, ensuring it extends in both directions. Remember, the slope tells the line's direction: rising, falling, or constant.
Algebra Concepts
In algebra, linear functions like \( f(x) = -3x \) require a strong grasp of basic concepts to manipulate and graph them effectively. Basics such as understanding coefficients, variables, and constants are keys to working with these equations.
  • **Variables** are symbols representing unknown values, often denoted as \( x \) and \( y \).
  • **Coefficients** are numbers attached to variables that show scaling; here, \(-3\) is the coefficient of \( x \).
  • **Constants** are fixed values, like the y-intercept \( b \) in the slope-intercept form, which can be \(0\) or any other number.
These concepts are not only essential for understanding how linear equations behave but also for solving them. Maintaining clarity on each component helps break down and solve equations efficiently, providing a stepping stone to more complex algebra problems.

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Most popular questions from this chapter

Find the inverse of the given function by using the "undoing process," and then verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). (Objective 4) $$f(x)=-4 x-3$$

Show that \((f \circ g)(x)=x\) and \((g \circ f)\) \((x)=x\) for each pair of functions. \(f(x)=-\frac{3}{4} x+\frac{1}{3}\) and \(g(x)=-\frac{4}{3} x+\frac{4}{9}\)

Translate each statement of variation into an equation, and use \(k\) as the constant of variation. \(y\) varies inversely as the square of \(x\).

Show that \((f \circ g)(x)=x\) and \((g \circ f)\) \((x)=x\) for each pair of functions. \(f(x)=-2 x\) and \(g(x)=-\frac{1}{2} x\)

Determine the indicated functional values. (Objective 2 ) If \(f(x)=-2 x-6\) and \(g(x)=3 x+10\), find \((f \circ g)(5)\) and \((g \circ f)(-3)\).
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