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Chapter 7: Problem 153

Construct the graph of the function defined by \(\mathrm{y}=3 \mathrm{x}-9\).

Short Answer

Expert verified
To construct the graph of \(y = 3x - 9\), find two points on the line by choosing two x-values and solving for the corresponding y-values. For example, with \(x_1 = 0\) and \(x_2 = 3\), we find the points \((0, -9)\) and \((3, 0)\). Plot these points on a Cartesian plane and draw a line through them to represent the graph of the function.

Step by step solution

01

Find two points on the line

Choose any two x-values and substitute them into the function y=3x-9, then solve for the corresponding y-values. Let's choose x鈧� = 0 and x鈧� = 3 for simplicity. For x鈧� = 0: y = 3(0) - 9 y = -9 So, the first point on the line is (0, -9). For x鈧� = 3: y = 3(3) - 9 y = 0 So, the second point on the line is (3, 0).
02

Plot the points on a Cartesian plane

Draw a Cartesian plane, with x-axis and y-axis. Locate and plot the points (0, -9) and (3, 0) on the plane. Remember that the first number in the coordinates, x, tells you how far to move horizontally (left or right) from the origin and the second number, y, tells you how far to move vertically (up or down) from the origin.
03

Draw the line representing the function

Using a straight edge, draw a line through the points (0, -9) and (3, 0), and extend this line beyond the points as needed in both directions. This line represents the graph of the function y = 3x - 9.

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Key Concepts

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

Plotting Points on Cartesian Plane
To understand functions and their graphs, it's essential to master plotting points on a Cartesian plane. The Cartesian plane, also known as the coordinate grid, is a two-dimensional platform divided by the x-axis (horizontal) and the y-axis (vertical), with the origin (0,0) at their intersection.

When plotting a point, the format \( (x, y) \) is used, where \( x \) and \( y \) represent the position relative to the origin. For the point \( (0, -9) \) on our plane, start at the origin, and because the x-value is 0, you do not move left or right. Then, move down 9 units because the y-value is -9. Similarly, for the point \( (3, 0) \) move right from the origin 3 units along the x-axis, and since the y-value is 0, you do not move up or down.

  • Identify the axes and the origin.
  • Locate the x-coordinate on the horizontal axis.
  • Move parallel to the y-axis to reach the y-coordinate.
  • Mark the point where this movement leads you.
Practicing this procedure will make plotting points a breeze and prepare you for graphing entire functions.
Linear Equation
A linear equation is one of the fundamental concepts in algebra and represents a straight line when graphed on a Cartesian plane. It's expressed in the form \( y = mx + b \) where \( m \) is the slope of the line, showing its steepness, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

In the equation \( y = 3x - 9 \) from the exercise, \( m \) equals 3, and \( b \) equals -9. Here, the slope of 3 indicates that for every unit increase in \( x \) the value of \( y \) increases by 3 units. The y-intercept at -9 informs us that the line crosses the y-axis at the point \( (0, -9) \).

Understanding the slope and y-intercept helps to quickly sketch a rough graph of the line without plotting multiple points, enhancing your efficiency in solving linear equations.
Function Graphing
Graphing is a powerful tool for visualizing the relationship between two variables. For linear functions, the graph is always a straight line. The process begins with plotting at least two points that satisfy the function's equation and then connecting these points with a straight edge.

In our example, after plotting the points \( (0, -9) \) and \( (3, 0) \), we draw a line through them which extends indefinitely in both directions. This line gives a visual representation of all the possible solutions to the equation \( y = 3x - 9 \).

  • Start with plotting two or more points that lie on the line defined by the function.
  • Check the linearity by ensuring these points align in a straight path.
  • Connect the points with a straight line for the graph of the function.
Function graphing not only aids in understanding the behavior of the function but also serves as a crucial step in solving more complex mathematical problems.

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Most popular questions from this chapter

Sketch the graph of the following functions: \(\begin{aligned}&\text { (a) } \mathrm{f}(\mathrm{x})=\left\\{\begin{array}{cc}\mathrm{x} \\ (1 / 2) \mathrm{x}-2\end{array}\right. & \begin{array}{l}\mathrm{x} \leq 4 \\\ \mathrm{x}>4\end{array} \\\&\text { (b) } \mathrm{f}(\mathrm{x})=|4 \mathrm{x}+9|\end{aligned}\) (c) \(\mathrm{f}(\mathrm{x})=|\mathrm{x}| /(\mathrm{x}+1)\) \(x \neq-1\)

Find the domain \(D\) and the range \(R\) of the function \((x, x /|x|)\)

If \(\mathrm{y}=\mathrm{f}(\mathrm{x})=\left(\mathrm{x}^{2}-2\right) /\left(\mathrm{x}^{2}+4\right)\) and \(\mathrm{x}=\mathrm{t}+1\), express \(\mathrm{y}\) as a function of \(\mathrm{t}\)

Find the inverse function of \(\mathrm{f}\) if \(\mathrm{y}=\mathrm{f}(\mathrm{x})=2 \sqrt{\left(9-\mathrm{x}^{2}\right) \text { and }}\) f has domain \(\\{x \mid-3 \leq x \leq 0\\}\) and range \(\\{y \mid 0 \leq y \leq 6\\}\).

If \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-\mathrm{x}-3, \mathrm{~g}(\mathrm{x})=\left(\mathrm{x}^{2}-1\right) /(\mathrm{x}+2)\), and \(\mathrm{h}(\mathrm{x})=\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x})\), find \(\mathrm{h}(2)\)
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