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Chapter 5: Problem 26

For the following exercises, find the intercepts of the functions. $$ g(n)=-2(3 n-1)(2 n+1) $$

Short Answer

Expert verified
X-intercepts: \( n = \frac{1}{3}, -\frac{1}{2} \); Y-intercept: \( g(n) = 2 \).

Step by step solution

01

Understand the Problem

We need to find the intercepts of the function \( g(n) = -2(3n - 1)(2n + 1) \). The intercepts are the points where the graph of the function crosses the axes.
02

Find the X-Intercepts

To find the x-intercepts, set \( g(n) = 0 \) and solve for \( n \). That gives us \(-2(3n - 1)(2n + 1) = 0\). Since \(-2\) is nonzero, we can ignore it. We are left with solving two equations:1. \(3n - 1 = 0\)2. \(2n + 1 = 0\)Solving these gives the x-intercepts.
03

Solve 3n - 1 = 0

Add 1 to both sides of the equation:\[3n = 1\]Then divide by 3:\[n = \frac{1}{3}\]
04

Solve 2n + 1 = 0

Subtract 1 from both sides of the equation:\[2n = -1\]Then divide by 2:\[n = -\frac{1}{2}\]
05

Find the Y-Intercept

To find the y-intercept, set \( n = 0 \) and calculate \( g(n) \):\[g(0) = -2(3(0) - 1)(2(0) + 1) = -2(-1)(1) = 2\] This gives the y-intercept.

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Key Concepts

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

X-Intercepts
The x-intercepts of a function are where the graph crosses the x-axis. This means the value of the function is zero at these points. To find them for the function \( g(n) = -2(3n - 1)(2n + 1) \), set the entire expression to zero: \[ -2(3n - 1)(2n + 1) = 0 \] Here, you can ignore the \(-2\) because it won't affect the equation being zero.
  • First, solve \( 3n - 1 = 0 \). Add 1 to both sides, giving \( 3n = 1 \). Then divide by 3: \( n = \frac{1}{3} \).
  • Next, solve \( 2n + 1 = 0 \). Subtract 1 from both sides, giving \( 2n = -1 \). Then divide by 2: \( n = -\frac{1}{2} \).
So, the x-intercepts are \( n = \frac{1}{3} \) and \( n = -\frac{1}{2} \). The graph will cross the x-axis at these points.
Y-Intercept
The y-intercept of a function is where the graph crosses the y-axis. This happens when the input, or \( n \), is zero. For \( g(n) = -2(3n - 1)(2n + 1) \), find the y-intercept by plugging \( n = 0 \) into the function: \[ g(0) = -2(3(0) - 1)(2(0) + 1) \] Simplify the expression:
  • \( 3(0) - 1 = -1 \)
  • \( 2(0) + 1 = 1 \)
  • Then, calculate \( -2(-1)(1) = 2 \)
Therefore, the y-intercept is 2. This tells us that the graph crosses the y-axis at the point (0, 2).
Solving Equations
An essential part of finding intercepts is solving equations. When the function is set to zero to find x-intercepts or when specific values are substituted to find a y-intercept, equation-solving skills are key. There are different types of equations you might encounter:
  • Linear equations like \( 3n - 1 = 0 \) or \( 2n + 1 = 0 \), which are solved by isolating \( n \).
When solving these, breakdown the steps into:
  • Isolating terms involving \( n \)
  • Simplifying using basic arithmetic operations (addition, subtraction, multiplication, division)
These skills allow you to determine intercepts accurately and handle more complex equations down the road.
Graph of the Function
Understanding the graph of a function is crucial in visualizing intercepts. The x-intercepts \( n = \frac{1}{3} \) and \( n = -\frac{1}{2} \) indicate points where the graph will touch or cross the x-axis. Meanwhile, the y-intercept shows where the graph crosses the y-axis, which for our function is at (0, 2). Graphically:
  • X-intercepts are points on the x-axis, visually identifiable as zero values in the function output.
  • The y-intercept is where the curve or line crosses the y-axis.
Plotting these on a coordinate plane helps in understanding the behavior of the graph, predicting how the curve behaves between and beyond these intercepts. This insight is fundamental for analyzing more complex functions effectively.

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