QuestionGiven the function $g(n)=(n-1)(n+6)(n-7)$ its $g$ -intercept is $\square$ its $n$ -intercepts are $\square$

Studdy Solution

STEP 1

What is this asking? We're looking for where this curvy function $g(n)$ crosses the vertical and horizontal axes! Watch out! Don't mix up the intercepts!
The $g$ -intercept is where $n=0$ , and the $n$ -intercepts are where $g(n)=0$ .

STEP 2

1. Find the $g$ -intercept.
2. Find the $n$ -intercepts.

STEP 3

To find the $g$ -intercept, we need to figure out the value of $g(n)$ when $n$ is zero.
This is where the function crosses the vertical axis.
So, let's substitute $n=0$ into our function: $g(0) = (0-1)(0+6)(0-7)$

STEP 4

Now, let's evaluate this expression: $g(0) = (-1)(6)(-7)$ $g(0) = (-1 \cdot 6) \cdot (-7)$ $g(0) = (-6) \cdot (-7)$ $g(0) = 42$ So, the $g$ -intercept is 42!

STEP 5

To find the $n$ -intercepts, we set the entire function $g(n)$ equal to zero.
This is where the function crosses the horizontal axis. $g(n) = (n-1)(n+6)(n-7) = 0$

STEP 6

This equation is already factored for us, how convenient!
A product of factors is zero if and only if at least one of the factors is zero.
So, we set each factor equal to zero and solve for $n$ .

STEP 7

$n-1 = 0$ We add one to both sides of the equation: $n - 1 + 1 = 0 + 1$ $n = 1$

STEP 8

$n+6 = 0$ We add negative six to both sides of the equation: $n + 6 - 6 = 0 - 6$ $n = -6$

STEP 9

$n-7 = 0$ We add seven to both sides of the equation: $n - 7 + 7 = 0 + 7$ $n = 7$

STEP 10

So, our $n$ -intercepts are $n=1$ , $n=-6$ , and $n=7$ .

STEP 11

The $g$ -intercept is 42.
The $n$ -intercepts are 1, -6, and 7.

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