Problem 21 - 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. $$ y=-x+4 $$

Step-by-Step Solution

Answer

The x-intercept is (4, 0) and the y-intercept is (0, 4); there is no symmetry.

Step 1: Create a Table of Values

To create a table of values, choose several x-values and calculate their corresponding y-values using the equation \( y = -x + 4 \). For example: \( x = -2, -1, 0, 1, 2 \).- If \( x = -2 \), then \( y = -(-2) + 4 = 6 \).- If \( x = -1 \), then \( y = -(-1) + 4 = 5 \).- If \( x = 0 \), then \( y = -(0) + 4 = 4 \).- If \( x = 1 \), then \( y = -(1) + 4 = 3 \).- If \( x = 2 \), then \( y = -(2) + 4 = 2 \). Record these in a table: \[\begin{array}{c|c}x & y \\hline-2 & 6 \-1 & 5 \0 & 4 \1 & 3 \2 & 2 \\end{array}\]

Step 2: Sketch the Graph

Plot the points from the table \((-2, 6), (-1, 5), (0, 4), (1, 3), (2, 2)\) on a coordinate plane. Since this is a linear equation, draw a straight line through the points. The line will slope downwards as the equation has a negative slope (-1).

Step 3: Find the x-intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \).\[0 = -x + 4 \x = 4\]So, the x-intercept is at \( (4, 0) \).

Step 4: Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( y \). From the equation: \( y = -(0) + 4 = 4 \). So, the y-intercept is at \( (0, 4) \).

Step 5: Test for Symmetry

For symmetry, test the equation for even and odd functions.- To test for y-axis symmetry, substitute \( x = -x \) and see if the equation remains the same. \(-x + 4 eq x + 4\), hence it is not symmetrical about the y-axis.- To test for x-axis symmetry, substitute \( y = -y \): leads to \(-y = -x + 4\), which simplifies back to \( y eq x - 4 \) and is not symmetrical about the x-axis.- To test origin symmetry, substitute \( (x,y) \to (-x,-y) \): \(-y = x + 4 \) which does not simplify to the original equation, so it is not symmetrical about the origin.

Key Concepts

Graphing Linear Equations

Graphing linear equations is about visualizing a line on a coordinate grid. When you have an equation like \( y = -x + 4 \), it represents a straight line. To graph this equation, start by creating a table of values. Choose several points for \( x \) and compute the corresponding \( y \) values by plugging them into the equation. For instance:

• If \( x = -2 \), \( y = (-(-2) + 4) = 6 \).
• If \( x = 0 \), \( y = (-(0) + 4) = 4 \).
• Continue with more values for accuracy.

After deriving the points, plot them on the graph and draw a line through them. Since this line has a slope of \(-1\), it will slope downward as you move from left to right. The concept is straightforward: the graph of any linear equation will always be a straight line.
Such plots are useful in many real-world applications, from economics to physics, where linear relationships often appear.

Intercepts of Graphs

Intercepts are specific points where the graph meets the axes. For linear equations, they help in sketching the graph accurately.
The **y-intercept** is where the graph intersects the y-axis. To find it, simply set \( x = 0 \) in the equation and solve for \( y \). In our case:

• y-intercept: \( (0, 4) \).

The **x-intercept** is where the graph crosses the x-axis. Here, set \( y = 0 \) and solve for \( x \). For the equation \( y = -x + 4 \):

• x-intercept: \( (4, 0) \).

These intercepts are key to getting a quick sketch of the graph.
They are particularly helpful for understanding where phenomena start or stop, such as when a company breaks even (x-intercept) or the initial amount in a bank account (y-intercept).

Symmetry in Graphs

Symmetry in graphs indicates whether, and how, the shape of the graph is mirrored over specific axes or points. Testing for symmetry can provide insights about the graph's behavior and properties. For symmetry:

• Check **y-axis symmetry** by replacing \( x \) with \( -x \). The line \(-x + 4 eq x + 4\) shows no y-axis symmetry.
• Check **x-axis symmetry** by replacing \( y \) with \( -y \). With \(-y = -x + 4 eq x - 4\), no x-axis symmetry exists.
• Check **origin symmetry** by substituting \( (x, y) \) with \( (-x, -y) \). The transformation \(-y = x + 4 eq y = -x + 4 \) results in no symmetry about the origin.

No symmetry simplifies sketching the graph, as it avoids the need to plot mirrored points. Understanding symmetry is crucial in higher-level mathematics and physics, especially in topics involving reflections and transformations.