Problem 57 Make a table of values and sketc... [FREE SOLUTION]

91影视

Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 1: Problem 57

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 the y-intercept is (0, 4); there's no symmetry.

Step by step solution

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 $y$.

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:\[\begin{array}{c|c} x & y \\hline -2 & 6 \ -1 & 5 \ 0 & 4 \ 1 & 3 \ 2 & 2 \\end{array}\]

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.

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 $(4, 0)$.

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 $(0, 4)$.

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 reflection properties, though for this particular linear function, symmetry isn't present. Knowing how to test symmetry simplifies interpreting more complex graphs.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Step by step solution

Create a Table of Values

Fill in the Table

Plot the Points on a Graph

Find the x-intercept

Find the y-intercept

Check for Symmetry

Key Concepts

Graphing

Intercepts

Symmetry

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.