/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Sketch a graph of the line. $$... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 8

Sketch a graph of the line. $$g(x)=-2 x-5$$

Short Answer

Expert verified
The graph of \(g(x) = -2x - 5\) is a straight line with a slope of -2 and it crosses the y-axis at \(y = -5\).

Step by step solution

01

Identify the slope and y-intercept

The standard form of a linear equation is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. From \(g(x) = -2x - 5\), we can identify that the slope, \(m\), is -2 and the y-intercept, \(c\), is -5.
02

Plot the y-intercept

Start by plotting the y-intercept on the graph. As \(c = -5\) here, put a point at \(y = -5\) on the y-axis.
03

Use the slope to find the next point

From the y-intercept, move according to the slope to find the next point. As the slope is -2 (which can be interpreted as -2/1), it means for every single unit you move to the right on the x-axis, you must move two units down on the y-axis. So, from your point at \(y = -5\), move one step to the right and two steps down, then put another point.
04

Draw the line

Now, draw a straight line through the two points. This line represents the graph of the equation \(g(x) = -2x - 5\).

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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 line in a linear equation is an essential concept that tells us how steep the line is. It is represented by the letter \( m \) in the equation \( y = mx + c \). The slope shows how much the value of \( y \) changes when \( x \) changes by one unit.
In our given equation \( g(x) = -2x - 5 \), the slope \( m \) is -2.
- A negative slope means the line is slanted downwards from left to right, indicating a decrease in \( y \) as \( x \) increases.
- It can be expressed as a fraction, i.e., \( -2/1 \), which means for every 1 unit you move to the right on the x-axis, you go 2 units down on the y-axis.
Understanding the slope helps in sketching the line on the graph, as it guides the direction and steepness of the line.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This point is crucial because it gives us a starting point for graphing a linear equation.
The y-intercept is denoted by \( c \) in the equation \( y = mx + c \).
In the equation \( g(x) = -2x - 5 \), the y-intercept \( c \) is -5.
- It means that the line crosses the y-axis at -5, hence the coordinate of this point is \( (0, -5) \).
- Graphically, it is the first point you plot when sketching a line.
By starting at this point and using the slope, we can accurately draw the line that represents the equation.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It consists of two perpendicular lines known as the x-axis and the y-axis.
Understanding the coordinate plane is vital for graphing linear equations.
- The x-axis runs horizontally and is used to determine the horizontal position of a point.
- The y-axis runs vertically and is used to find the vertical position of a point.
When plotting the equation \( g(x) = -2x - 5 \) on the coordinate plane, we can use the identified y-intercept and apply the slope to find more points on the line.
- Begin by plotting the y-intercept at (0, -5) on the y-axis.
- Use the slope to find another point by moving right 1 unit and down 2 units, marking this point around the graph.
Connecting these points with a straight line helps us visualize the relationship represented by the linear equation on the coordinate plane.

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Most popular questions from this chapter

Solve the inequality. Express your answer in interval notation. $$1 \leq \frac{2 x-1}{3} \leq 4$$

Graph the function by hand. $$f(x)=\left\\{\begin{array}{ll} -2, & x<-1 \\ |x|, & -1 \leq x \leq 2 \\ 2, & 2

Explain why the following table of function values cannot be that of a linear function. $$\begin{array}{|c|c|} \hline \hline t & g(t) \\ \hline -2 & -3 \\ -1 & -5 \\ 0 & -8 \\ 1 & -15 \\\ \hline \end{array}$$

This set of exercises will draw on the ideas presented in this section and your general math background. Let \(f\) be defined as followos. $$ f(x)=\left\\{\begin{array}{ll} 0, & \text { if } x \leq 1 \\ 2, & \text { if } x>1 \end{array}\right. $$ Graph \(f(x-1)\)

Let \(g(t)=m t+b .\) Find \(m\) and \(b\) such that \(g(1)=4\) and \(g(3)=4 .\) Write an expression for \(g(t) .\) (Hint: Start by using the given information to write down the coordinates of two points that satisfy \(g(t)=m t+b .\) )
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