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Chapter 0: Problem 13

Sketch the graphs of each pair of functions on the same coordinate plane. $$ y=x, y=-x $$.

Short Answer

Expert verified
Plot points for each function and draw the corresponding lines.

Step by step solution

01

- Understand the Functions

The functions given are: 1. y = x 2. y = -x. They represent two different types of linear equations.
02

- Identify the Slopes and Intercepts

For y = x, the slope is 1 and the y-intercept is 0. For y = -x, the slope is -1 and the y-intercept is also 0.
03

- Plot Points for y = x

Choose a few values for x and find the corresponding y values for y = x. For instance, if x = -2, y = -2, if x = 0, y = 0, and if x = 2, y = 2. Plot these points: (-2,-2), (0,0), and (2,2).
04

- Plot Points for y = -x

Choose a few values for x and find the corresponding y values for y = -x. For example if x = -2, y = 2, if x = 0, y = 0, and if x = 2, y = -2. Plot these points: (-2, 2), (0, 0), and (2, -2).
05

- Draw the Lines

Draw a straight line through the points plotted in step 3 for y = x. Similarly, draw another straight line through the points plotted in step 4 for y = -x. These are the graphs of the two functions on the same coordinate plane.

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Key Concepts

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

coordinate plane
When we talk about graphing linear equations, the coordinate plane is our playground. It's like a big piece of graph paper with horizontal and vertical lines.
The horizontal line is called the x-axis. The vertical line is called the y-axis.
Where these two lines intersect is called the origin, which is at the point (0,0).
This setup allows us to plot points and draw lines to represent equations.
linear equations
Linear equations are equations that make a straight line when you graph them. They have the general form y = mx + b, where 'm' is the slope, and 'b' is the y-intercept.
In simpler terms, the slope tells us how steep the line is, and the y-intercept is where the line crosses the y-axis.
For example, y = x is a simple linear equation with a slope of 1 and a y-intercept of 0. On the other hand, y = -x has a slope of -1 and also crosses the y-axis at 0.
slope-intercept form
The slope-intercept form is a handy way to write linear equations. The formula is y = mx + b.
The 'm' in the equation represents the slope, which tells us how much y changes for each change in x.
The 'b' represents the y-intercept. For instance, in the equation y = x, the slope (m) is 1 and the y-intercept (b) is 0.
This means the line goes up one unit for every one unit it goes to the right. In y = -x, the slope is -1, meaning the line goes down one unit for every one unit it moves to the right.
plotting points
Plotting points is like connecting the dots to draw a picture. You start with a pair of numbers called coordinates (x, y).
For example, to plot the point (2, 3), you start at the origin (0,0). Then move 2 units to the right (along the x-axis) and 3 units up (along the y-axis).
To graph each equation, you plot enough points and then connect them to draw the line.
For y = x, you might plot points like (1,1), (2,2), and (-1,-1) and draw a line through them. For y = -x, you could plot (1,-1), (2,-2), and (-1,1), and then draw a line through those points.

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