x y

To find the equation that describes the relationship in the table, we can identify the pattern from the given \( x \) and \( y \) values.

We have the following pairs:

• When \( x = 0 \), \( y = -10 \)
• When \( x = 1 \), \( y = -9 \)
• When \( x = 2 \), \( y = -8 \)
• When \( x = 3 \), \( y = -7 \)

From the data, we can observe that as \( x \) increases by 1, \( y \) increases by 1. This indicates a linear relationship between \( x \) and \( y \).

To express this in slope-intercept form \( f(x) = mx + b \):

1. Find the slope (m): Each time \( x \) increases by 1, \( y \) increases by 1. Thus, \( m = 1 \).

2. Find the y-intercept (b): We can see from the table that when \( x = 0 \), \( y = -10 \). Therefore, \( b = -10 \).

Putting it all together, we have the equation:

\[ f(x) = 1x - 10 \]

So, the equation described in the table is:

\[ f(x) = x - 10 \]

The number of bugs doubles every 4 days. If the bug population starts with 6 bugs, then the equation A(t)=6(2)^0.25t
can represent the number of bugs. Find the value for A(8)
The number of bugs after 8 days is (___)

To find the value for \( A(8) \) using the equation \( A(t) = 6(2)^{0.25t} \), we will substitute \( t = 8 \) into the equation:

\[ A(8) = 6(2)^{0.25 \cdot 8} \]

Calculating \( 0.25 \cdot 8 \):

\[ 0.25 \cdot 8 = 2 \]

Now substitute this back into the equation:

\[ A(8) = 6(2)^2 \]

Calculating \( (2)^2 \):

\[ (2)^2 = 4 \]

So now we can find \( A(8) \):

\[ A(8) = 6 \cdot 4 = 24 \]

The number of bugs after 8 days is 24.

The distance away from the city of Bloomsfield is a function of time. The function D(h)=440−65h
represents the distance in miles away from Bloomsfield as you travel toward the city, based on the number of hours, h , you have traveled. Find the distance left to get to Bloomsfield after 3 1/2
hours of traveling. Enter your response in decimal form.
The distance that you have left to travel is (___)
miles.

To find the distance left to get to Bloomsfield after \( 3.5 \) hours of traveling using the function \( D(h) = 440 - 65h \), we need to substitute \( h = 3.5 \) into the equation.

\[ D(3.5) = 440 - 65 \cdot 3.5 \]

Now let's calculate \( 65 \cdot 3.5 \):

\[ 65 \cdot 3.5 = 227.5 \]

Next, substitute this value back into the equation:

\[ D(3.5) = 440 - 227.5 \]

Subtract \( 227.5 \) from \( 440 \):

\[ D(3.5) = 440 - 227.5 = 212.5 \]

So, the distance left to travel to Bloomsfield after \( 3.5 \) hours is 212.5 miles.

x y
0 −5

1 −9

2 −13

3 −17

Which of the following equations correctly describes the function in the table using function notation?



(1 point)
Responses

f(x)=−5x−4

f of x is equal to negative 5 x minus 4

f(y)=−4x−5
f left parenthesis y right parenthesis equals negative 4 x minus 5

f(y)=−5x−4
f left parenthesis y right parenthesis equals negative 5 x minus 4

f(x)=−4x−5

To find the correct equation that describes the function in the table, let's analyze the given points:

• When \( x = 0 \), \( y = -5 \)
• When \( x = 1 \), \( y = -9 \)
• When \( x = 2 \), \( y = -13 \)
• When \( x = 3 \), \( y = -17 \)

Step 1: Find the pattern.

Looking at the \( y \) values, we can observe that as \( x \) increases by 1, \( y \) decreases by 4.

Step 2: Determine the slope (m).

The slope between the points can be calculated as follows: \[ \text{slope} = \frac{\Delta y}{\Delta x} = \frac{-9 - (-5)}{1 - 0} = \frac{-4}{1} = -4 \]

Step 3: Use the slope-intercept form \( y = mx + b \).

Now that we have the slope \( m = -4 \) and we know that \( y = -5 \) when \( x = 0\), we can find the y-intercept \( b \):

Taking the point \( (0, -5) \): \[ y = mx + b \Rightarrow -5 = -4(0) + b \Rightarrow b = -5 \]

Step 4: Write the equation.

Putting it together, the equation is: \[ y = -4x - 5 \]

In function notation, that is: \[ f(x) = -4x - 5 \]

Conclusion:

The correct response is: \[ f(x) = -4x - 5 \]

Evaluate the function for the following input: f(x)=3−2x
. Which input generates the largest output when evaluating the function?(1 point)
Responses

f(2)
f left parenthesis 2 right parenthesis

f(3)
f left parenthesis 3 right parenthesis

f(5)
f left parenthesis 5 right parenthesis

f(0)

To evaluate the function \( f(x) = 3 - 2x \) for the given inputs and determine which produces the largest output, we will calculate \( f(2) \), \( f(3) \), \( f(5) \), and \( f(0) \).

1. Calculate \( f(2) \): \[ f(2) = 3 - 2(2) = 3 - 4 = -1 \]

2. Calculate \( f(3) \): \[ f(3) = 3 - 2(3) = 3 - 6 = -3 \]

3. Calculate \( f(5) \): \[ f(5) = 3 - 2(5) = 3 - 10 = -7 \]

4. Calculate \( f(0) \): \[ f(0) = 3 - 2(0) = 3 - 0 = 3 \]

Now we compare the outputs:

• \( f(2) = -1 \)
• \( f(3) = -3 \)
• \( f(5) = -7 \)
• \( f(0) = 3 \)

The largest output is from \( f(0) = 3 \).

Thus, the input that generates the largest output is:

\(f(0)\).

The function V(r)=10πr^2
represents the volume of a cylinder that is 10 inches high with a radius of r. Which is the correct interpretation for the function notation V(3)=90π
?(1 point)
Responses

The volume of the cylinder is 90π
when the radius is 3.
The volume of the cylinder is 90 pi when the radius is 3.

The volume of the cylinder is 3 when the radius is 90π
.
The volume of the cylinder is 3 when the radius is 90 pi .

A cylinder of 90π
is produced from a radius of 3.
A cylinder of 90 pi is produced from a radius of 3.

A radius of 3 gives the value 90.

To interpret the function notation \( V(3) = 90\pi \) in the context of the function \( V(r) = 10\pi r^2 \), we need to consider what the function represents:

• The function \( V(r) \) calculates the volume of a cylinder with a fixed height of 10 inches and a variable radius \( r \).
• The value \( V(3) = 90\pi \) means that when the radius \( r \) is 3 inches, the volume of the cylinder is \( 90\pi \) cubic inches.

Based on this understanding, the correct interpretation of the function notation \( V(3) = 90\pi \) is:

The volume of the cylinder is \( 90\pi \) when the radius is 3.