Problem 24 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ x+y=3 $$

Step-by-Step Solution

Answer

The x-intercept is (3, 0) and the y-intercept is (0, 3). The graph is a line with negative slope, without symmetry about axes.

Step 1: Understanding the Equation

The given equation is \(x + y = 3\). This is a linear equation in two variables, \(x\) and \(y\). Our task is to find intercepts, test for symmetry, and plot the graph.

Step 2: Finding the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\).\[\begin{align*} x + 0 &= 3 \ x &= 3 \end{align*}\]Thus, the x-intercept is at \((3, 0)\).

Step 3: Finding the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\).\[\begin{align*} 0 + y &= 3 \ y &= 3 \end{align*}\]Thus, the y-intercept is at \((0, 3)\).

Step 4: Making a Table of Values

Let's create a table of values by choosing different values for \(x\) and calculating corresponding \(y\) values using \(y = 3 - x\).\[\begin{array}{c|c} x & y \\hline 0 & 3 \ 1 & 2 \ 2 & 1 \ 3 & 0 \ 4 & -1 \\end{array}\]

Step 5: Sketching the Graph

Plot the points from the table: \((0, 3), (1, 2), (2, 1), (3, 0), (4, -1)\). Connect the dots with a straight line. This is the graph of \(x + y = 3\), a straight line with a slope of -1.

Step 6: Testing for Symmetry

A linear equation of the form \(Ax + By = C\) is only symmetric about the origin when both \(A\) and \(B\) are non-zero. Since \(x + y = 3\) doesn't change sign upon replacing \(x\) or \(y\) with their negatives, it is only symmetric with respect to itself, having no symmetry about axes or origin.

Key Concepts

X-Intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the y-coordinate is zero. To find the x-intercept of the equation \(x + y = 3\), we substitute \(y = 0\) into the equation. Now the equation becomes \(x + 0 = 3\), which simplifies to \(x = 3\). So, the x-intercept is the point \((3, 0)\). This means that the graph will cross the x-axis at the point \( (3, 0) \). Identifying x-intercepts helps in understanding the starting or ending point of a graph along the x-axis.

Y-Intercept

The y-intercept of a graph indicates the point where the graph crosses the y-axis. At this intersecting point, the x-coordinate is zero. For the equation \(x + y = 3\), the y-intercept can be found by substituting \(x = 0\) into the equation. This results in \(0 + y = 3\), which simplifies to \(y = 3\). Thus, the y-intercept is the point \((0, 3)\). This point signifies where the line meets the y-axis, and it is crucial for accurately sketching the graph, as it provides a point from which the line can extend.

Graphing Linear Equations

Graphing linear equations like \(x+y=3\) involves plotting points that satisfy the equation, then drawing a straight line through those points.

• First, select a set of x-values. For each value, compute the corresponding y-value using the equation.
• For instance, choosing x-values such as 0, 1, 2, 3, 4 gives us the points: \((0,3), (1,2), (2,1), (3,0), (4,-1)\).
• Plot these points on a graph, ensuring they are accurately represented.
• Finally, draw a straight line through them, which in this case will have a negative slope, indicating it tilts downward as you move from left to right.

Graphing provides a visual understanding of linear equations, highlighting features such as intercepts and the line's slope. Consistent pattern in plotting might even reveal additional intersecting points.

Symmetry in Graphs

Symmetry in graphs is an aspect that reveals how a graph mirrors itself across axes. For linear equations like \(x + y = 3\), the concept of symmetry helps us understand how the graph behaves when coordinates are altered.

• Type of symmetry depends on whether the equation maintains its form when variables are replaced with their negative values.
• A line symmetric about the y-axis remains unchanged if every x is replaced by -x.
• However, being a linear equation, \(x + y = 3\) shows no symmetry about the x or y axes for it doesn't satisfy the criteria for axis or origin symmetry.

Symmetry testing enhances understanding of equations by confirming whether they maintain consistency across transformations, thus offering insights into more complex graph behaviors.