Problem 19

Question

Sketch the graph of $f$. $f(x, y)=x+2 y$

Step-by-Step Solution

Verified

Answer

The graph of $f(x, y) = x + 2y$ is a plane in 3D space, with intercepts on the z-axis and steady slopes in x and y directions.

1Step 1: Understand the Function

The given function is $ f(x, y) = x + 2y $. This is a linear equation in two variables, $x$ and $y$. The graph of such a function is a plane in three-dimensional space.

2Step 2: Create a Table of Values

Choose specific values for $x$ and $y$, and calculate the corresponding $f(x, y)$ values.For example:- If $x = 0$ and $y = 0$, then $f(x, y) = 0 + 2(0) = 0$.- If $x = 1$ and $y = 0$, then $f(x, y) = 1 + 2(0) = 1$.- If $x = 0$ and $y = 1$, then $f(x, y) = 0 + 2(1) = 2$.

3Step 3: Identify the Intercepts

To sketch the graph, find the intercepts:- **x-intercept**: Set $y = 0$ in the equation, giving $f(x, 0) = x + 2(0) = x$. The x-intercept is at $(x, 0, x)$.- **y-intercept**: Set $x = 0$ in the equation, giving $f(0, y) = 0 + 2y = 2y$. The y-intercept is at $(0, y, 2y)$.

4Step 4: Determine the Plane

The equation $z = x + 2y$ represents a plane in 3D space. From Step 3, we know that this plane passes through the z-axis at the origin (0, 0, 0), and for every unit movement in x, the z-coordinate increases by 1, while for every unit movement in y, the z-coordinate increases by 2.

5Step 5: Sketch the Graph

On a 3D coordinate system, plot the points based on the values obtained. Connect the points to form a plane. You can choose three or more non-collinear points to fully draw the plane. Use the intercepts and the principle of linear increase (1 unit in $x$ direction and 2 units in $y$ direction) to guide your sketch.

Key Concepts

Linear Equations in Two VariablesIntercepts in GraphsPlanes in 3D Space

Linear Equations in Two Variables

Linear equations in two variables are fundamental concepts in mathematics. They are equations that can be expressed in the form: $ ax + by = c $, where $ a $, $ b $, and $ c $ are constants. In these equations, $ x $ and $ y $ are variables that can take on various values.

When you graph a linear equation with two variables on a two-dimensional plane, you'll observe a straight line. The line represents all possible solutions for the equation. Linear equations are popular because they are simple, easy to calculate, and their solutions provide insights into relationships between variables.

An interesting aspect of these equations is that their graph extends indefinitely in both directions. This is because there are infinitely many combinations of $ x $ and $ y $ that satisfy the equation. Linear equations in two variables represent the most basic relationship between two quantities, yet they are versatile in representing data in various fields such as economics, physics, and engineering.

Intercepts in Graphs

When graphing functions, intercepts are crucial points that provide significant information about the graph's behavior. Intercepts are where the graph crosses the axes.

X-intercept: Occurs where the graph crosses the x-axis. To find it, set $ y = 0 $ in the equation and solve for $ x $.
Y-intercept: Occurs where the graph crosses the y-axis. For this, set $ x = 0 $ in the equation and solve for $ y $.

These intercepts aid in sketching the graph as they offer reference points around which the rest of the graph is structured. Take, for example, a linear equation in two variables like $ f(x, y) = x + 2y $. The x-intercept happens when $ y = 0 $, giving us $ (x, 0, x) $ in 3D space. The y-intercept occurs when $ x = 0 $, yielding $ (0, y, 2y) $ in 3D space.

Intercepts simplify the plotting process since you already have key coordinates on the graph. When these intercepts are plotted accurately, they form the basis for drawing a meaningful and correct representation of the equation.

Planes in 3D Space

Graphs of linear equations that include an expression like $ f(x, y) = x + 2y $ are represented as planes in 3D space. This is a step beyond typical line graphs in 2D, as we now add an extra dimension: the z-dimension.

A plane in 3D space can be visualized as a flat, infinitely extending surface. Any linear equation of the form $ z = ax + by + c $ graphs as a plane. Here, $ a $ and $ b $ influence the orientation and slant of the plane, while $ c $ affects its position relative to the origin.

For the equation $ z = x + 2y $, steps to determine the characteristics of the plane include:

Identifying its passage through the origin (0, 0, 0) where, for any given $ x $ and $ y $, $ z $ is calculated using the equation.
Recognizing the rate of change in the z-value: for every unit increase in $ x $, $ z $ increases by 1; for every unit increase in $ y $, $ z $ increases by 2.

Using two or more points based on calculated values, the plane is visualized and drawn on a 3D graph. Understanding how planes intersect with axes and other planes builds the groundwork for more advanced geometry and calculus applications.

Problem 19

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