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Chapter 15: Problem 57

Sketch the graphs of the functions. HINT [See Example 7.] $$ g(x, y)=2 x+y-2 $$

Short Answer

Expert verified
To sketch the graph of the function \(g(x, y) = 2x + y - 2\), rewrite it as \(y = -2x + 2 + g(x, y)\), and identify its slope (-2) and intercepts. To sketch the graph, plot the x-intercept (\(y = 2 + g(0, y)\)) and y-intercept (\(x = 0\)). Use the slope, -2, to plot additional points on the line by moving downwards 2 units for every unit to the right. Finally, draw a straight line connecting all the points to represent the graph of the function.

Step by step solution

01

Rewriting the equation

First, let's rewrite the given function in the slope-intercept form by solving for y: \(g(x, y) = 2x + y - 2\). Rewrite it as: $$ y = -2x + 2 + g(x, y) $$ So, we have the linear equation of y in terms of x and g(x,y).
02

Identifying the slope and intercepts

From the equation \(y = -2x + 2 + g(x, y)\), we can identify the slope and the intercepts: The slope of the linear function is -2 (the coefficient of x). Therefore, the line will have a negative slope, meaning it will be going downward from left to right. The y-intercept is when x = 0. In this case, the equation becomes: $$ y = -2(0) + 2 + g(0, y) = 2 + g(0, y) $$ The x-intercept is when y = 0. In this case, the equation becomes: $$ 0 = -2x + 2 + g(x, 0) $$ Now we have all the information needed for sketching the graph.
03

Sketching the graph

To sketch the graph of the function, follow these steps: a. Start by plotting the x and y intercepts. We have \(x = 0\) for the y-intercept, and \(y = 2 + g(0, y)\) for the x-intercept. b. Next, use the slope of the line, -2, to plot additional points on the line. Since the slope is -2, the line will go down 2 units for every 1 unit to the right. You can plot other points on the graph by moving from the x and y intercepts. c. Finally, draw a straight line connecting all the points plotted. This line will be the graph of the function \(g(x, y) = 2x + y - 2\). Note that in this process, we didn't actually plot the points onto the graph, because it is not necessary to plot the graph of g(x,y) on the 3D axis. We only needed to sketch the graph of the linear function on the 2D x-y plane.

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form is pivotal when working with linear equations. This form is expressed as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept. This form makes it easy to visualize and sketch the graph of a linear equation quickly.

- **Slope \( (m) \)**: The slope is a measure of how steep the line is. In our equation \( y = -2x + 2 \), the slope \( m \) is -2. This tells us that for every unit we move to the right along the x-axis, the line goes down 2 units. This represents a negative slope.

- **Y-Intercept \( (b) \)**: The y-intercept is where the line crosses the y-axis. From our equation, \( y \) crosses the y-axis at 2, i.e., when \( x = 0 \).

Using the slope-intercept form, you can quickly draw the line by starting at the y-intercept \( (0, 2) \) and using the slope to determine the direction and angle of the line.
X-Intercept
The x-intercept is an essential component for understanding any linear equation. It is the point where the graph crosses the x-axis.

To find the x-intercept, set \( y = 0 \) in your equation, because at the point where the line crosses the x-axis, the y-value is zero. Using the equation \( y = -2x + 2 \), replace \( y \) with 0:\[0 = -2x + 2\]
Solve for \( x \) by isolating \( x \) on one side of the equation:
\[2x = 2\]\[x = 1\]
So, the line crosses the x-axis at the point \( (1, 0) \). This gives a concrete starting or reference point for graphing the line.
Y-Intercept
The y-intercept of a linear equation is the point at which the graph crosses the y-axis. This point has its significance as it allows us to start sketching our graph directly from this value.

In the slope-intercept form \( y = mx + b \), the y-intercept is denoted by \( b \). For the equation \( y = -2x + 2 \), \( b = 2 \). Thus, the line crosses the y-axis at the point \( (0, 2) \).

Finding the y-intercept is simple:
  • Substitute \( x = 0 \) in the equation.
  • Solve for \( y \).
Since the y-intercept is \( (0, 2) \), start plotting your graph at this point, then use the slope to determine the line's direction. The intercept is a crucial anchor for drawing accurate linear graphs.

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Most popular questions from this chapter

Use Lagrange multipliers to solve the given optimization problem. HINT [See Example 2.] Find the maximum value of \(f(x, y)=x y\) subject to \(3 x+y=\) 60 . Also find the corresponding point(s) \((x, y)\).

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The Gym Shirt Company manufactures cotton socks. Production is partially automated through the use of robots. Daily operating costs amount to $$\$ 150$$ per laborer and $$\$ 60$$ per robot. The number of pairs of socks the company can manufacture in a day is given by a Cobb-Douglas production formula $$ q=50 n^{0.6} r^{0.4} $$ where \(q\) is the number of pairs of socks that can be manufactured by \(n\) laborers and \(r\) robots. Assuming that the company has a daily operating budget of $$\$ 1,500$$ and wishes to maximize productivity, how many laborers and how many robots should it use? What is the productivity at these levels? HINT [See Example 5.]
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