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Chapter 6: Problem 2

A linear function is given. Determine the \(x\) -intercept and \(y-\) intercept, and then graph the linear function. If the function does not have an \(x\) -intercept, then say so. $$f(x)=12-3 x$$

Short Answer

Expert verified
X-intercept: (4, 0); Y-intercept: (0, 12); Graph is a line through these points.

Step by step solution

01

Identify the Linear Function

The given linear function is \( f(x) = 12 - 3x \). This function is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Determine the Y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( f(x) \).\[ f(0) = 12 - 3(0) = 12\] Thus, the y-intercept is \( (0, 12) \).
03

Find the X-intercept

To find the x-intercept, set \( f(x) = 0 \) and solve for \( x \).\[ 0 = 12 - 3x \] Solving this equation:\[3x = 12 \]\[x = 4 \]Thus, the x-intercept is \( (4, 0) \).
04

Graph the Linear Function

To graph the function, plot the x-intercept \((4, 0)\) and the y-intercept \((0, 12)\) on the coordinate plane. Since it's a linear function, draw a straight line through these two intercepts.

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Key Concepts

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

x-intercept
The concept of the x-intercept is essential when dealing with linear equations. It is the point where the graph of the function crosses the x-axis. For any point on the x-axis, the y-coordinate is always zero. Therefore, to find the x-intercept, we set the function equal to zero and solve for x. In the linear function \( f(x) = 12 - 3x \), this means:
  • Set \( f(x) = 0 \) 鈫� \( 12 - 3x = 0 \).
  • Rearrange to solve for x 鈫� \( 3x = 12 \).
  • Divide by 3 鈫� \( x = 4 \).
Thus, the x-intercept is the point \( (4, 0) \). This tells us the graph will cross the x-axis at x = 4.
Understanding the x-intercept can help predict where the graph will intersect the x-axis and analyze the behavior of the function.
y-intercept
The y-intercept is another critical concept in graphing linear functions. This is where the graph intersects the y-axis, and at this point, the x-coordinate is zero. Finding the y-intercept of a linear function like \( f(x) = 12 - 3x \) involves substituting zero for x:
  • Substitute x = 0 into \( f(x) \) 鈫� \( f(0) = 12 - 3(0) \).
  • Simplify 鈫� \( f(0) = 12 \).
Thus, the y-intercept is at \( (0, 12) \). This point reveals that when x is zero, the function value is 12, marking where the line crosses the y-axis.
Grasping the y-intercept is particularly handy in analyzing how the graph of a function starts or shifts along the y-axis.
graphing linear equations
Graphing linear equations involves drawing the line that represents the function on a coordinate plane. The linear equation \( f(x) = 12 - 3x \) is in slope-intercept form \( y = mx + b \), where the graph of the equation is a straight line. The slope \( m \) determines the steepness, and the y-intercept \( b \) shows where the line crosses the y-axis.
To graph this function:
  • Use the x-intercept \( (4, 0) \) and the y-intercept \( (0, 12) \) as the key points.
  • Plot both intercepts on a graph.
  • Draw a straight line through these points.
This process results in a visual representation of the function. Graphing can help in understanding the relationship between variables and visually inspecting the slope and intercepts. It allows you to predict values for unmeasured data points and detect patterns or trends in real-world data.

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