Math  /  Algebra

QuestionDetermine the domain on which the following function is decreasing.

Studdy Solution

STEP 1

What is this asking? Where does the graph of this parabola go down as we move from left to right? Watch out! Remember, we're looking for where the graph is going *downhill* from left to right, not where the *y*-values are negative!

STEP 2

1. Find the vertex.
2. Determine the decreasing interval.

STEP 3

Alright, let's **locate the vertex**!
The vertex of a parabola is like the tip-top of a mountain (if it opens downwards) or the bottom of a valley (if it opens upwards).
In this case, our parabola opens downwards, so we're looking for the highest point.

STEP 4

Looking at the graph, we can see that the **vertex** is at the point (2,9)(-2, 9).
This means the *x*-coordinate of the vertex is 2-2 and the *y*-coordinate is 99.
Remember, the *x*-coordinate of the vertex is super important for figuring out where the function increases or decreases!

STEP 5

Now that we know the **vertex**, we can figure out where the function is decreasing.
Since our parabola opens downwards, the function decreases to the *right* of the vertex.

STEP 6

The *x*-value of our **vertex** is 2-2.
So, the function is decreasing for all *x*-values *greater* than 2-2.
We can write this **interval** using interval notation as (2,)(-2, \infty).
The parenthesis around 2-2 means that 2-2 itself is *not* included in the interval, and the infinity symbol means that the interval goes on forever in the positive direction.

STEP 7

The function is decreasing on the interval (2,)(-2, \infty).

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