Problem 85 Graph by using the slope and \(y... [FREE SOLUTION]

Chapter 12: Problem 85

Graph by using the slope and $y$ -intercept. (GRAPH CAN'T COPY) $$y=-x+1$$

Short Answer

Expert verified

The graph of the linear equation $y = -x + 1$ is a downward straight line that passes through the points $(0, 1)$ and $(1, 0)$.

Step by step solution

Identify and plot the y-intercept

The y-intercept of the given equation is $1$. Therefore, the first point to plot on the graph is $(0, 1)$, indicating that the line crosses the y-axis at the point where $y = 1$. This point can be plotted by starting at the origin and moving one unit in the positive direction along the y-axis.

Use the slope to find another point

The slope of the equation is $-1$, which can be interpreted as the ratio of the vertical change ($\Delta y$) to the horizontal change ($\Delta x$). A slope of $-1$ means that for each unit moved in the positive x-direction, we move one unit in the negative y-direction. Starting from the y-intercept $(0, 1)$, moving one unit to the right and one unit down gives us the coordinate $(1, 0)$. This is the second point we will plot.

Draw the line

With two points, $(0, 1)$ and $(1, 0)$, identified on the graph, draw a straight line through these points to represent the graph of the equation. This line indicates all possible points $(x, y)$ that satisfy the equation $y = -x + 1$.

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Key Concepts

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

Slope-Intercept Form

The slope-intercept form is a very handy way to express the equation of a straight line. It provides all the information you need to graph a line quickly. The general format is:\[ y = mx + b \]Here:

$ y $ is the dependent variable.
$ x $ is the independent variable.
$ m $ represents the slope of the line.
$ b $ is the y-intercept, which is where the line crosses the y-axis.

This format simplifies the process of graphing because it directly gives you the slope and the y-intercept. That's why it is often preferred when you are asked to graph a linear equation.
For example, in the equation $ y = -x + 1 $, the slope $ m $ is $-1$ and the y-intercept $ b $ is $ 1 $. With this information, you can instantly plot the y-intercept and use the slope to find other points on the line.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. This point occurs where the x-coordinate is zero. It is a crucial point because it's the starting point for graphing our line using the slope.
In the slope-intercept form $ y = mx + b $, the $ b $ represents the y-intercept. So, for the equation $ y = -x + 1 $, the y-intercept is the point $ (0, 1) $.
To plot the y-intercept:

Start at the origin $(0,0)$.
Move up or down along the y-axis to the y-intercept valued at $ b $.
In this case, move up 1 unit to $(0,1)$.

Once you have plotted the y-intercept, you can use the slope to find additional points and establish the line's direction on the graph.

Slope

The slope of a line describes how steep the line is and the direction it goes. It's calculated as the change in the y-values divided by the change in the x-values, often referred to as 'rise over run.'
In the equation $ y = mx + b $, the $ m $ represents the slope. For $ y = -x + 1 $, the slope $ m $ is $-1$. This tells us two main things:

The line decreases as you move from left to right because the slope is negative.
For every one unit you move to the right along the x-axis, you move one unit down along the y-axis.

When graphing, start at the y-intercept and use the slope to find the next point.
From $(0, 1)$, move 1 unit to the right and 1 unit down to arrive at $(1, 0)$.This systematic approach ensures that your line accurately represents the equation every time.

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91影视

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$y=-x+1$$

Short Answer

Step by step solution

Identify and plot the y-intercept

Use the slope to find another point

Draw the line

Key Concepts

Slope-Intercept Form

Y-Intercept

Slope

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