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Magazine Chapter 1: Problem 51
Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts and test for symmetry. $$y=-x+4$$
Short Answer
Expert verified
The x-intercept is (4,0) and y-intercept is (0,4); no symmetry about the axes or origin.
Step by step solution
01
Choose Values for x
To create a table of values for the equation \(y = -x + 4\), select several values of \(x\). Commonly, choosing values like -2, -1, 0, 1, and 2 will provide a good range to see the behavior of the line.
02
Calculate Corresponding y Values
Substitute each \(x\) value chosen in Step 1 into the equation \(y = -x + 4\) and solve for \(y\). For example: \(x = -2\) gives \(y = -(-2) + 4 = 6\). Repeat for each \(x\) value.
03
Create a Table of Values
List the chosen \(x\) values and their corresponding \(y\) values from Step 2 in a table. For example:- \((-2, 6)\)- \((-1, 5)\)- \((0, 4)\)- \((1, 3)\)- \((2, 2)\)
04
Plot Points and Sketch the Graph
On a coordinate plane, plot the five points from your table. Connect these points with a straight line because the equation is linear, implying a straight-line graph.
05
Find the x-intercept
To find the \(x\)-intercept, set \(y = 0\) in the equation \(y = -x + 4\) and solve for \(x\). Solving \(0 = -x + 4\) gives \(x = 4\), so the \(x\)-intercept is \((4, 0)\).
06
Find the y-intercept
To find the \(y\)-intercept, set \(x = 0\) in the equation \(y = -x + 4\) and solve for \(y\). Solving \(y = -0 + 4\) gives \(y = 4\), so the \(y\)-intercept is \((0, 4)\).
07
Test for Symmetry
Check for symmetry about the y-axis, x-axis, or origin.- Replace \(x\) with \(-x\) in the equation and check if the result is the same as the original equation for y-axis symmetry.- The equation \(y = -x + 4\) becomes \(y = x + 4\), which is not equal to the original, so it is not symmetric about the y-axis.- Not symmetric about the x-axis or origin based on visual inspection and typical symmetry tests for linear equations.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
X-Intercept Fundamentals
The x-intercept is the point where a graph crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept of the linear equation \( y = -x + 4 \), follow these simple steps.
- Set \( y = 0 \) in the equation. This is because the x-intercept happens precisely where the graph meets the x-axis, leaving y out of the picture.
- Solve the equation \( 0 = -x + 4 \) for x. By adding x to both sides or subtracting 4 from both sides, you'll find \( x = 4 \).
- This means the x-intercept is at the point \( (4, 0) \).
Y-Intercept Insight
The y-intercept is similar to the x-intercept, but it is where the graph crosses the y-axis. Here, the x-coordinate is always zero. To find the y-intercept of the equation \( y = -x + 4 \), there鈥檚 an easy process to follow.
- Set \( x = 0 \) because the y-intercept occurs on the y-axis where x is zero.
- This turns the equation into \( y = -0 + 4 \), which simplifies to \( y = 4 \).
- Hence, the y-intercept is located at the point \( (0, 4) \).
Understanding Symmetry in Linear Equations
Symmetry refers to the balance and harmony in the shape of a graph. For linear equations, symmetry can be examined around the y-axis, the x-axis, or even the origin. But for many linear equations like \( y = -x + 4 \), symmetry is rarely present.
- To test for y-axis symmetry, replace \( x \) with \(-x\) in the equation.
- If the new equation is exactly the same as the original after simplification, the graph is symmetric about the y-axis.
- For our equation: replacing \( x \) with \( -x \) gives \( y = x + 4 \).
- Since \( y = x + 4 \) does not match the original \( y = -x + 4 \), it shows no y-axis symmetry.
- Visual inspection and symmetry tests indicate that \( y = -x + 4 \) is not symmetric about the x-axis or the origin either.
