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Chapter 0: Problem 32

Graph each function. $$ g(x)=-2 x $$

Short Answer

Expert verified
The graph is a straight line passing through the origin (0,0) with a slope of -2.

Step by step solution

01

Identify the Type of Function

The function given is a linear function because it can be written in the form of \(g(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this function, \(g(x) = -2x\), the slope \(m = -2\) and the y-intercept \(b = 0\).
02

Plot the Y-Intercept

Since the y-intercept \(b = 0\), the graph of the function passes through the origin (0, 0). Plot the point (0, 0) on the graph.
03

Use the Slope to Find Another Point

The slope \(m = -2\) means that for every 1 unit you move to the right along the x-axis, you move 2 units down along the y-axis (since it is negative). Starting from the origin (0, 0), move 1 unit to the right to (1, 0), then 2 units down to (1, -2). Plot the point (1, -2).
04

Draw the Line

Using the two points (0, 0) and (1, -2) that you have plotted, draw a straight line through these points, extending it in both directions. This line represents the graph of the function \(g(x) = -2x\).

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Key Concepts

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

Linear Equations
Linear equations are mathematical expressions that describe a straight line on a graph. The general form of a linear equation is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. These equations are simple because they only involve operations of addition, subtraction, and multiplication. They do not contain exponents or complex functions. This simplicity allows them to model many real-world scenarios.
  • Consist of two variables, typically \(x\) and \(y\).
  • Graph as a straight line on a coordinate plane.
  • Used to find unknown values when given the relationship between variables.
In the function \(g(x) = -2x\), the linear equation describes a line that passes through the origin with a specific slope but no constant term, making the y-intercept zero.
Slope
The slope of a line is a crucial concept in understanding linear equations. It indicates how steep a line is and the direction it moves on a graph. The slope is calculated as the change in \(y\) over the change in \(x\) (often referred to as rise over run). For the given function \(g(x) = -2x\), the slope \(m\) is \(-2\).
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope, like \(-2\), means the line falls in the same direction.
  • A slope of zero would indicate a horizontal line.
With a slope of \(-2\), for every 1 unit you move to the right along the x-axis, you descend 2 units on the y-axis. This movement directly affects the appearance of the graph, making it look like it's going downwards from left to right.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis. It provides a specific location on the graph, often making it easier to plot the line. In a linear equation \(y = mx + b\), the y-intercept is the value of \(b\). For the function \(g(x) = -2x\), the y-intercept \(b\) is 0.
  • Indicates the starting point of the graph on the y-axis.
  • When \(b = 0\), the line passes through the origin \((0, 0)\).
  • Helps in quickly sketching the graph by providing a known point.
Because \(g(x) = -2x\) passes through the origin, this means the y-intercept is at \((0, 0)\), serving as one of the two main points needed to draw the line.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and curves. It is defined by a horizontal axis (x-axis) and a vertical axis (y-axis) intersecting at a point called the origin \((0, 0)\).
  • Used to plot graphs of equations, including linear equations.
  • Divided into four quadrants by the x and y axes.
  • Helps visualize the relationship between variables.
In the case of our linear function \(g(x) = -2x\), plotting starts at the coordinate \((0, 0)\), and because the slope is \(-2\), you follow the rule of moving down 2 units and over 1 unit to the right to plot additional points. This movement helps define the line in this plane.

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Most popular questions from this chapter

Rewrite each of the following as an equivalent expression with rational exponents. $$ \sqrt{x^{5}} $$

Rewrite each of the following as an equivalent expression using radical notation. $$ t^{2 / 5} $$

An anthropologist can use certain linear functions to estimate the height of a male or female, given the length of certain bones. The humerus is the bone from the elbow to the shoulder. Let \(x=\) the length of the humerus, in centimeters. Then the height, in centimeters, of a male with a humerus of length \(x\) is given by $$ M(x)=2.89 x+70.64 $$. The height, in centimeters, of a female with a humerus of length \(x\) is given by $$ F(x)=2.75 x+71.48$$ A \(26-\mathrm{cm}\) humerus was uncovered in some ruins. a) If we assume it was from a male, how tall was he? b) If we assume it was from a female, how tall was she?

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In general, people in our society are marrying at a later age. The median age, \(A(t),\) of women at first marriage can be approximated by $$ A(t)=0.08 t+19.7 $$, where \(t\) is the number of years after \(1950 .\) Thus, \(A(0)\) is the median age of women at first marriage in 1950 \(A(50)\) is the median age in \(2000,\) and so on. a) Find \(A(0), A(1), A(10), A(30),\) and \(A(50)\) b) What was the median age of women at first marriage in \(2008 ?\) c) Graph \(A(t)\)
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