Give the formula for the following sequence. 22, 19, 16, 13, 10,...

x_n = x_(n-1) - 3

Step-by-step explanation:

x_n = x_(n-1) - 3

Take the previous number and subtract 3

The graph points down when x is large and negative, up when x is large and positive.

The y intercept is at x = -18.

The zeros are at x = -3, -2, and 3.

Step-by-step explanation:

Hey, Kelsey. Three key features of a function g(x) are the end behavior, the y-intercept, and the zeros.

Let's look at your function g(x) = x³ + 2x² − 9x − 18 and try to discover these features.

1. The end behaviour

The end behaviour of the function is what happens to the function as x ⟶ ∞ and as x ⟶ -∞.

The important point for you to remember is that, when x is large enough, the term of highest degree (the x³ term) will be so large that you can ignore all the other terms.

If x is large and negative, say, -100,

g(x) ≈ (-100)³ = -1 000 000 = -1 000 000  

Thus, the graph is in the third quadrant and it points down.

If x is large and positive, say, 100,

g(x) ≈ (100)³ = 1 000 000 = 1 000 000  

Thus, the graph is in the first quadrant and it points up.

2. The y-Intercept

The y-intercept is  the value of g(x) when x = 0.

That's easy to calculate. You just insert zero into the expression. Then,

g(0) = 0³ + 2(0)² − 9(0) − 18 = 0 + 2(0) - 0 -18  = -18

3. The zeros

The zeros are the roots of the equation, that is, the values of x that make g(x) = 0.

That means you must solve the equation

x³ + 2x² − 9x − 18 = 0

You must try to factor the cubic equation. That is not always easy, and sometimes it's impossible.

In this, case, you can factor by grouping.

Group the terms into groups of two and factor each one separately.

x³ + 2x² − 9x − 18 = (x³ + 2x²) − (9x − 18) = x²(x + 2) - 9(x + 2)

Now you can use the distributive property of x + 2 to write

g(x) = (x² - 9)(x + 2) = 0

Next, you can use the zero-product property which states that a product of factors is zero if one or more of the factors is zero. So,

x + 2 = 0     x² - 9 = 0

The first factor is easy to solve.

x = -2

The second factor is a difference of two squares. It factors as  

x² - 9 = (x + 3)(x - 3) = 0

Again, you can use the zero-product property:

x + 3 = 0     x - 3 = 0

      x = -3         x = 3

You have found all three zeroes of g(x): -3, -2, and 3.