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Chapter 3: Problem 66

Explain how to graph an equation written in point-slope form.

Short Answer

Expert verified
To graph an equation in point-slope form, identify the slope and point from the equation, plot the point, use the slope to identify another point and plot it, then draw a line through the points.

Step by step solution

01

Identify the slope and the point

Identify the slope and point using the formula of point-slope form. The coefficient in front of \( (x - x_1) \) in the equation is the slope, and the values for \( x_1 \) and \( y_1 \) give the coordinates of the point.
02

Plot the identified point

Plot the point \( (x_1, y_1) \) on the graph. This is a point that we know lies on the line.
03

Graph the line using the slope

Starting from the plotted point, use the slope to identify another point on the line and plot it. The slope, \( m \), is a ratio representing the vertical change (also called the 'rise') over the horizontal change (the 'run'). If the slope is positive, go up and to the right from the starting point; if it's negative, go down and to the right.
04

Draw the line

After plotting at least two points, draw a straight line through the points. This is the graph of the equation.

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Key Concepts

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

Graphing Linear Equations
When graphing linear equations, the primary task is to represent the equation as a straight line on a graph. Each linear equation can be written in several forms, with the point-slope form being particularly useful for graphing. This form is expressed as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \((x_1, y_1)\) is a known point on the line.

Linear equations generally describe a constant rate of change, which is depicted as a straight line on a coordinate plane. To graph such an equation effectively:
  • Identify key elements like slope and specific points.
  • Understand how these elements interact with the axes of the graph.
  • Use them to create a visual representation of the equation.
By plotting linear equations, you can visually analyze relationships and patterns that are evident in algebraic expressions.
Algebraic Graphing Techniques
Algebraic graphing techniques are essential tools in transforming equations from paper to graph. One popular technique involves using the point-slope form for easy visualization. This form gives a straightforward method to plot linear equations as it incorporates a specific point and the slope, providing an easy start and direction for your graph.

To enhance your graphing skills:
  • Start by plotting a known point using its coordinates \((x_1, y_1)\).
  • Utilize the slope \(m\) to find additional points. The slope \(m\) describes how much \(y\) changes as \(x\) changes.
  • Draw the line by connecting points as instructed, ensuring a clear and accurate representation of the linear equation.
This methodical approach ensures precise graphing and helps you better understand the relationship between algebraic equations and their graphical representations.
Slope and Coordinates
The slope and coordinates are crucial elements in graphing linear equations. The slope, usually denoted by \(m\), quantifies the steepness and direction of a line on a graph. It is calculated as the ratio of the vertical change ('rise') to the horizontal change ('run') between two points on a line.

Understanding slope:
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves left to right.
  • A slope of zero indicates a horizontal line.
  • An undefined slope is typical of a vertical line.
Coordinates are used to specify a particular point on the graph with the format \((x, y)\). In the point-slope form, a specific point \((x_1, y_1)\) is used as a starting location to apply the slope. By understanding how slope and coordinates interact, you can accurately plot any linear equation in point-slope form effectively on a graph.

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Most popular questions from this chapter

Set up a variation equation and solve for the requested value. The interest earned on a fixed amount of money varies jointly with the annual interest rate and the time that the money is left on deposit. If an account earns \(120\)dollar at \(8 \%\) annual interest when left on deposit for 2 years, how much interest would be earned in 3 years at an annual rate of \(12 \% ?\)

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. \(A(8,-6)\) and the origin

Set up a variation equation and solve for the requested value. The time it takes to build a highway varies directly with the length of the highway and inversely with the number of workers. If it takes 100 workers 4 weeks to build 2 miles of highway, how long will it take 80 workers to build 10 miles of highway?

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. $$P(3,8) \text { and } Q(9,-2)$$

Express each inverse variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 2. (OBJECTIVE 2) \(r\) varies inversely with \(s .\) If \(r=40\) when \(s=10,\) find \(r\) when \(s=15\).
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