Math Problem Statement

Given the function g(n)=(n-2)(n+5)(n-3):

the coordinates of its g-intercept are 

the coordinates of its n-intercepts are 

Solution

Let's analyze the given function g(n)=(n2)(n+5)(n3)g(n) = (n - 2)(n + 5)(n - 3) to find its intercepts.

1. g-intercept (y-intercept):

The g-intercept occurs when n=0n = 0. So, substitute n=0n = 0 into the function:

g(0)=(02)(0+5)(03)=(2)(5)(3)=30g(0) = (0 - 2)(0 + 5)(0 - 3) = (-2)(5)(-3) = 30

Thus, the coordinates of the g-intercept are:

(0,30)(0, 30)

2. n-intercepts (x-intercepts):

The n-intercepts occur where g(n)=0g(n) = 0. Set the function equal to 0:

(n2)(n+5)(n3)=0(n - 2)(n + 5)(n - 3) = 0

This equation is satisfied when any of the factors equals zero:

  • n2=0n=2n - 2 = 0 \Rightarrow n = 2
  • n+5=0n=5n + 5 = 0 \Rightarrow n = -5
  • n3=0n=3n - 3 = 0 \Rightarrow n = 3

Thus, the coordinates of the n-intercepts are:

(2,0),(5,0),(3,0)(2, 0), (-5, 0), (3, 0)

Summary:

  • g-intercept: (0,30)(0, 30)
  • n-intercepts: (2,0),(5,0),(3,0)(2, 0), (-5, 0), (3, 0)

Do you have any questions or want further details?

Here are some related questions to consider:

  1. How do you find the vertex of a quadratic or cubic function?
  2. What does the sign of the leading coefficient tell you about the graph of the function?
  3. How can you use the factored form of a polynomial to sketch its graph?
  4. What is the significance of multiple roots in a polynomial function?
  5. How does the degree of a polynomial function affect its general shape?

Tip: When finding intercepts, always plug n=0n = 0 for the g-intercept and set the function equal to zero for the n-intercepts.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Factoring
Intercepts

Formulas

g(n) = (n - 2)(n + 5)(n - 3)
n-intercept: g(n) = 0
g-intercept: g(0)

Theorems

Zero Product Property
Factored Form of Polynomials

Suitable Grade Level

Grades 9-11