Math / Calculus

QuestionEssay 25 points Explain how you calculated the answer to the previous question number 6. Show your work using the formula necessary to calculate AROC.
What is the average rate of change between inputs 1 to 5? Edit View Insert Format Tools Table 12pt Paragraph B I U A 2 $T^{2}$ $0^{\circ}$ 斻

Studdy Solution

STEP 1

What is this asking? We need to find the average rate of change of the graphed function between $x = 1$ and $x = 5$ . Watch out! Don't mix up the $x$ and $y$ values when using the formula.
Also, remember that the average rate of change is different from the instantaneous rate of change.

STEP 2

1. Find the function values
2. Calculate the average rate of change

STEP 3

Alright, let's kick things off by finding the value of the function at $x = 1$ .
Looking at the graph, we see that when $x = 1$ , the function value is approximately $y = 3$ .
So, we have the point $(1, 3)$ .
Remember, this first point's coordinates are $(x_1, y_1)$ , so $x_1 = 1$ and $y_1 = 3$ .

STEP 4

Now, let's do the same thing for $x = 5$ .
From the graph, when $x = 5$ , the function value is approximately $y = -2$ .
This gives us the point $(5, -2)$ .
This second point's coordinates are $(x_2, y_2)$ , so $x_2 = 5$ and $y_2 = -2$ .

STEP 5

The magic formula for the average rate of change (AROC) is: $\text{AROC} = \frac{y_2 - y_1}{x_2 - x_1}$ This formula tells us how much the function's output ( $y$ ) changes on average for every unit change in the input ( $x$ ).
It's like finding the slope of the line connecting our two points!

STEP 6

Let's plug in our values.
Remember, we found $x_1 = 1$ , $y_1 = 3$ , $x_2 = 5$ , and $y_2 = -2$ .
Substituting these values into the AROC formula, we get: $\text{AROC} = \frac{-2 - 3}{5 - 1}$

STEP 7

Time to simplify! In the numerator, $-2 - 3 = -5$ .
In the denominator, $5 - 1 = 4$ .
So, we have: $\text{AROC} = \frac{-5}{4}$

STEP 8

We can write this as a decimal: $\text{AROC} = -1.25$

STEP 9

The average rate of change between $x = 1$ and $x = 5$ is $-1.25$ .
This tells us that, on average, for every one unit increase in $x$ , the function's value decreases by $1.25$ units.

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Studdy Solution

STEP 1

What is this asking? We need to find the average rate of change of the graphed function between $x = 1$ and $x = 5$ . Watch out! Don't mix up the $x$ and $y$ values when using the formula.
Also, remember that the average rate of change is different from the instantaneous rate of change.

STEP 2

1. Find the function values
2. Calculate the average rate of change

STEP 3

Alright, let's kick things off by finding the value of the function at $x = 1$ .
Looking at the graph, we see that when $x = 1$ , the function value is approximately $y = 3$ .
So, we have the point $(1, 3)$ .
Remember, this first point's coordinates are $(x_1, y_1)$ , so $x_1 = 1$ and $y_1 = 3$ .

STEP 4

Now, let's do the same thing for $x = 5$ .
From the graph, when $x = 5$ , the function value is approximately $y = -2$ .
This gives us the point $(5, -2)$ .
This second point's coordinates are $(x_2, y_2)$ , so $x_2 = 5$ and $y_2 = -2$ .

STEP 5

The magic formula for the average rate of change (AROC) is: $\text{AROC} = \frac{y_2 - y_1}{x_2 - x_1}$ This formula tells us how much the function's output ( $y$ ) changes on average for every unit change in the input ( $x$ ).
It's like finding the slope of the line connecting our two points!

STEP 6

Let's plug in our values.
Remember, we found $x_1 = 1$ , $y_1 = 3$ , $x_2 = 5$ , and $y_2 = -2$ .
Substituting these values into the AROC formula, we get: $\text{AROC} = \frac{-2 - 3}{5 - 1}$

STEP 7

Time to simplify! In the numerator, $-2 - 3 = -5$ .
In the denominator, $5 - 1 = 4$ .
So, we have: $\text{AROC} = \frac{-5}{4}$

STEP 8

We can write this as a decimal: $\text{AROC} = -1.25$

STEP 9

The average rate of change between $x = 1$ and $x = 5$ is $-1.25$ .
This tells us that, on average, for every one unit increase in $x$ , the function's value decreases by $1.25$ units.

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Studdy Solution

STEP 1

What is this asking? We need to find the average rate of change of the graphed function between x=1x = 1x=1 and x=5x = 5x=5. Watch out! Don't mix up the xxx and yyy values when using the formula.Also, remember that the *average* rate of change is different from the *instantaneous* rate of change.

STEP 2

1. Find the function values2. Calculate the average rate of change

STEP 3

STEP 4

STEP 5

STEP 6

Let's **plug in** our values.Remember, we found x1=1x_1 = 1x1​=1, y1=3y_1 = 3y1​=3, x2=5x_2 = 5x2​=5, and y2=−2y_2 = -2y2​=−2.Substituting these values into the AROC formula, we get: AROC=−2−35−1 \text{AROC} = \frac{-2 - 3}{5 - 1} AROC=5−1−2−3​

STEP 7

**Time to simplify!** In the numerator, −2−3=−5-2 - 3 = -5−2−3=−5.In the denominator, 5−1=45 - 1 = 45−1=4.So, we have: AROC=−54 \text{AROC} = \frac{-5}{4} AROC=4−5​

STEP 8

We can write this as a decimal: AROC=−1.25 \text{AROC} = -1.25 AROC=−1.25

STEP 9

The average rate of change between x=1x = 1x=1 and x=5x = 5x=5 is −1.25-1.25−1.25.This tells us that, on average, for every one unit increase in xxx, the function's value *decreases* by 1.251.251.25 units.

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What is this asking? We need to find the average rate of change of the graphed function between $x = 1$ and $x = 5$ . Watch out! Don't mix up the $x$ and $y$ values when using the formula.
Also, remember that the average rate of change is different from the instantaneous rate of change.

1. Find the function values
2. Calculate the average rate of change

Let's plug in our values.
Remember, we found $x_1 = 1$ , $y_1 = 3$ , $x_2 = 5$ , and $y_2 = -2$ .
Substituting these values into the AROC formula, we get: $\text{AROC} = \frac{-2 - 3}{5 - 1}$

Time to simplify! In the numerator, $-2 - 3 = -5$ .
In the denominator, $5 - 1 = 4$ .
So, we have: $\text{AROC} = \frac{-5}{4}$

We can write this as a decimal: $\text{AROC} = -1.25$

The average rate of change between $x = 1$ and $x = 5$ is $-1.25$ .
This tells us that, on average, for every one unit increase in $x$ , the function's value decreases by $1.25$ units.