Problem 77 Graph by using the slope and \(y... [FREE SOLUTION]

Chapter 12: Problem 77

Graph by using the slope and $y$ -intercept. (GRAPH CAN'T COPY) $$y=\frac{2}{5} x-2$$

Short Answer

Expert verified

To create the graph of $y=\frac{2}{5} x-2$, first plot a point at the y-intercept $-2$. Then, use the slope of $\frac{2}{5}$ to move up 2 and to the right 5 from the y-intercept, plot a point, and draw a line through the points. Label the line with the equation.

Step by step solution

Plot the Y-Intercept

Start by plotting a point at the y-intercept, which is $-2$ in this case. So, put a dot at the point where $y = -2$ on the y-axis.

Apply the Slope

From the y-intercept, apply the slope to find another point on the line. The slope is $\frac{2}{5}$, so move 2 units upward and 5 units to the right for each unit of x from the y-intercept.

Draw the Line

With the y-intercept plotted and using the slope to find another point on the line, you can then draw a straight line through these points. This line represents the solutions to the equation.

Label the Line

Finally, label the line with the equation $y=\frac{2}{5} x-2$ to indicate that it is the graph of this equation.

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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 an essential concept when dealing with linear equations. It takes the format of $y = mx + b$. Here, $m$ represents the slope of the line, while $b$ denotes the y-intercept. This form allows for easy graphing of linear equations as it directly provides both the slope and y-intercept, two critical components for plotting a line.

To use this form effectively:

Look at your equation and identify the slope $m$ and the y-intercept $b$.
Immediately place a point at the y-intercept on the y-axis.
Utilize the slope to determine how the line ascends or descends from that point.

These straightforward steps make the slope-intercept form particularly powerful for quick graph visualization. Graphing becomes a step-by-step exercise in plotting and connecting points for an accurate representation of the equation.

Y-intercept

The y-intercept is the point where the line crosses the y-axis on a graph. In the slope-intercept form $y = mx + b$, the $b$ is the y-intercept. This tells you exactly where to place the starting point of your line on the graph, specifically where the value of $x$ is zero, and the value of $y$ corresponds to $b$.

For the equation $y = \frac{2}{5}x - 2$, the y-intercept is $-2$, indicating that on the graph, the line crosses the y-axis at the point $ (0, -2) $.

Here's how you can use it:

Locate $b$ in the equation to determine the y-intercept.
Go to the y-axis on your graph and locate $b$.
Mark this point as one of the first critical points for your line.

This step anchors your graph, providing a fixed point from which to determine the line's direction with the slope.

Slope

The slope of a line in an equation is a measure of its steepness. It is represented by $m$ in the slope-intercept form $y = mx + b$. The slope tells you how much the line rises or falls as you move along the x-axis. It is calculated as the "rise" (change in $y$) over the "run" (change in $x$).

For the equation $y = \frac{2}{5}x - 2$, the slope is $\frac{2}{5}$. This means for every 5 units you move to the right along the x-axis, you move up by 2 units along the y-axis. To plot using the slope:

Start at the y-intercept point.
From this point, move right along the x-axis according to the denominator of the slope. For $\frac{2}{5}$, move 5 units to the right.
Move vertically according to the numerator. In this case, move 2 units up.

These directions help you find another point on the line, making it easier to draw and visualize the line accurately.

Plotting points

Plotting points is a fundamental part of graphing linear equations, helping to visualize and ensure accuracy of the linear representation. When you have both your slope and y-intercept, you're ready to locate and draw the main points of the graph.

Here's how you can plot points effectively:

Begin with the y-intercept and mark it on the graph.
Use the slope to identify at least one more point on the line. Move according to the slope from the y-intercept to locate this second point.
Mark this second point on the graph.
Once you have at least two points, simply draw a straight line through them.

By following these steps, you can accurately depict the line represented by an equation like $y = \frac{2}{5}x - 2$. Plotting points allows for a clear visual understanding of the relationship between the variables in a linear equation.

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91影视

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$y=\frac{2}{5} x-2$$

Short Answer

Step by step solution

Plot the Y-Intercept

Apply the Slope

Draw the Line

Label the Line

Key Concepts

Slope-intercept form

Y-intercept

Slope

Plotting points

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