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Chapter 2: Problem 35

Graph each equation using the intercept method. Label the intercepts on each graph. \(3 y=6 x-9\)

Short Answer

Expert verified
The intercepts are \((0, -3)\) and \(\left(\frac{3}{2}, 0\right)\).

Step by step solution

01

Convert to Slope-Intercept Form

The given equation is \(3y = 6x - 9\). First, we need to convert this equation into the standard form \(Ax + By = C\). To do this, divide every term by 3 to make \(y\) the subject: \[y = 2x - 3\]. Now the equation is in the form \(y = mx + b\), which is a slope-intercept form.
02

Find the Y-Intercept

In the equation \(y = 2x - 3\), the y-intercept is the constant term \(-3\). This means when \(x = 0\), \(y = -3\). The y-intercept is the point \((0, -3)\).
03

Find the X-Intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\): \[0 = 2x - 3\]. Adding 3 to both sides, we get \(2x = 3\). Now, divide both sides by 2: \(x = \frac{3}{2}\). The x-intercept is \(\left(\frac{3}{2}, 0\right)\).
04

Plot the Intercepts on the Graph

To graph the equation, plot the intercept points \((0, -3)\) and \(\left(\frac{3}{2}, 0\right)\) on a coordinate plane. These points are where the line crosses the y-axis and x-axis respectively. Draw a straight line through these two points to complete 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.

Intercept Method
The intercept method is a straightforward way to graph linear equations. It involves finding both the x-intercept and the y-intercept of the equation. These are the points where the graph of the equation crosses the x-axis and y-axis.
  • The x-intercept is found by setting the y-value to zero and solving for x.
  • The y-intercept is discovered by setting the x-value to zero and solving for y.
Once these intercepts are identified, plot them on a graph. Simply draw a straight line through both points to visualize the linear equation. This method is particularly useful for quickly sketching graphs and understanding the relationship in a linear equation.
Slope-Intercept Form
The slope-intercept form is a way of expressing the equation of a straight line. It provides a quick view of the line鈥檚 slope and y-intercept.The general form is: \[ y = mx + b \]
  • Here, \( m \) represents the slope or the steepness of the line.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
Converting an equation to slope-intercept form involves isolating y on one side of the equation. This form is convenient for graphing, as it directly tells you both the direction (slope) and starting point (y-intercept) of the line on a graph.
X-Intercept
The x-intercept is a critical feature of a graph where the line meets the x-axis. At this point, the value of y is zero.To find the x-intercept:
  • Set the equation鈥檚 y-variable to 0.
  • Solve for x, which gives you the exact point on the x-axis.
In our example, substituting \( y = 0 \) into the equation derived as \( 2x - 3 = 0 \) allows one to find \( x = \frac{3}{2} \). The point \( \left( \frac{3}{2}, 0 \right) \) is where our line crosses the x-axis. This point is essential for employing the intercept method.
Y-Intercept
The y-intercept is another key point of the graph of a line, indicating where the line crosses the y-axis. At this point, the x-value is inherently zero.To determine the y-intercept:
  • Set the x-variable of the equation to 0.
  • Solve for y, which gives you the y-intercept of the graph.
In the slope-intercept form, the y-intercept is directly given by the constant term. For example, in the equation \( y = 2x - 3 \), the y-intercept is \(-3\), corresponding to the point \( (0, -3) \). This helps you start graphing immediately by providing a specific location on the y-axis.

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Most popular questions from this chapter

Complete table. \(f(r)=-2 r^{2}+1\) \(\begin{array}{|r|l|}\hline \text { Input } & \text { Output } \\\\\hline-1.7 & \\\0.9 & \\\5.4 & \\\\\hline\end{array}\)

Let \(f(x)=-2 x+5 .\) For what value of \(x\) does function \(f\) have the given value? \(f(x)=-7\)

Write an equation for a linear function whose graph has the given characteristics. Passes through \((1,2),\) perpendicular to the graph of \(g(x)=-\frac{x}{6}+1\)

Explain why 8 is not in the domain of the function \(f(x)=\frac{5 x-7}{x-8}\).

Explain why we can think of a function as a machine.
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