Problem 57 - Precalculus Mathematics for Calculus

Question

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's no symmetry.

Step 1: Create a Table of Values

Choose values for \(x\) and calculate the corresponding \(y\) values using the equation \(y = -x + 4\). Fill in a table with these values. For example, let \(x = -2, -1, 0, 1, 2\) and calculate the values of \(y\).

Step 2: Fill in the Table

Calculate the corresponding \(y\) values for each \(x\):- For \(x = -2\), \(y = -(-2) + 4 = 6\)- For \(x = -1\), \(y = -(-1) + 4 = 5\)- For \(x = 0\), \(y = -(0) + 4 = 4\)- For \(x = 1\), \(y = -(1) + 4 = 3\)- For \(x = 2\), \(y = -(2) + 4 = 2\)The table is:\[\begin{array}{c|c} x & y \\hline -2 & 6 \ -1 & 5 \ 0 & 4 \ 1 & 3 \ 2 & 2 \\end{array}\]

Step 3: Plot the Points on a Graph

Use the values from the table to plot the points \((-2, 6)\), \((-1, 5)\), \((0, 4)\), \((1, 3)\), and \((2, 2)\) on a coordinate plane. Connect these points with a straight line, as the equation represents a linear function.

Step 4: Find the x-intercept

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

Step 5: Find the y-intercept

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

Step 6: Check for Symmetry

For symmetry: - **Y-axis Symmetry**: Replace \(x\) with \(-x\). \(y = -(-x) + 4 = x + 4\), which is not equivalent to the original equation.- **X-axis Symmetry**: Replace \(y\) with \(-y\). \(-y = -x + 4\), implying \(y = x - 4\), which is not equivalent.- **Origin Symmetry**: Replace \(x\) with \(-x\) and \(y\) with \(-y\). \(-y = -(-x) + 4 = x + 4\), which is not equivalent to the original.The function has no symmetry.

Key Concepts

Graphing

Graphing a linear equation like \(y = -x + 4\) involves several straightforward steps. Start by creating a table of values. Pick some simple x-values, calculate the corresponding y-values using the equation, and list them. Here’s how to do it:

• Choose values for \(x\), such as \(-2, -1, 0, 1, 2\).
• For each \(x\), compute \(y\) using \(y = -x + 4\).

Once you have your points (e.g., \((-2, 6), (-1, 5), (0, 4), (1, 3), (2, 2)\)), mark them on a Cartesian plane. Join these points with a straight line, confirming they align (since it's a linear function). This plotted straight line is the visual representation of your equation, revealing both direction and spacing.

Intercepts

The intercepts of a graph provide where the line crosses the axes. These points can be determined easily with a linear equation. Let's explore how:

• **X-intercept**: Set \(y = 0\) and solve for \(x\). For our function, \(0 = -x + 4\). Solving this, \(x = 4\). Thus, the x-intercept is \((4, 0)\).
• **Y-intercept**: Set \(x = 0\) and solve for \(y\). Substituting gives \(y = -0 + 4\), resulting in \(y = 4\). Therefore, the y-intercept is \((0, 4)\).

Finding intercepts is crucial because it gives precise contact points of the line with the axes. These insights can sometimes simplify plotting, offering a sense of the line's path even without a detailed graph.

Symmetry

Symmetry in graphs helps identify a function's visual balance. With linear functions, assessing symmetry involves replacement tests on the equation. For \(y = -x + 4\), examine as follows:

• **Y-axis Symmetry**: Replace \(x\) with \(-x\). You get \(y = x + 4\), which doesn't match the original. Hence, there is no y-axis symmetry.
• **X-axis Symmetry**: Substitute \(y\) with \(-y\). This results in \(-y = -x + 4\), leading to \(y = x - 4\), differing from the original. There is no x-axis symmetry.
• **Origin Symmetry**: Substitute \(x\) with \(-x\) and \(y\) with \(-y\). It simplifies to \(-y = x + 4\), dissimilar to the initial function, showing a lack of origin symmetry.

Understanding symmetry gives insights into the graph’s reflection properties, though for this particular linear function, symmetry isn't present. Knowing how to test symmetry simplifies interpreting more complex graphs.