Which of the following points does not belong to the graph of the function y=(x+1)^3−2

Question

?(1 point)
Responses

(−2,−3)
left parenthesis negative 2 comma negative 3 right parenthesis

(−3,−10)
left parenthesis negative 3 comma negative 10 right parenthesis

(0,−1)
left parenthesis 0 comma negative 1 right parenthesis

(−3,6)

Answer 1

To determine which point does not belong to the graph of the function \( y = (x+1)^3 - 2 \), we can substitute the \( x \)-coordinate of each point into the function and check if the resulting \( y \)-value matches the \( y \)-coordinate of that point.

Let’s evaluate each point one by one:

1. For the point \((-2, -3)\):

   • \( x = -2 \)
   • \( y = (-2 + 1)^3 - 2 = (-1)^3 - 2 = -1 - 2 = -3 \)
   • This point is on the graph.

2. For the point \((-3, -10)\):

   • \( x = -3 \)
   • \( y = (-3 + 1)^3 - 2 = (-2)^3 - 2 = -8 - 2 = -10 \)
   • This point is on the graph.

3. For the point \((0, -1)\):

   • \( x = 0 \)
   • \( y = (0 + 1)^3 - 2 = (1)^3 - 2 = 1 - 2 = -1 \)
   • This point is on the graph.

4. For the point \((-3, 6)\):

   • \( x = -3 \)
   • Calculate \( y = (-3 + 1)^3 - 2 = (-2)^3 - 2 = -8 - 2 = -10 \)
   • But this point has \( y = 6 \), which does not match.

The point \((-3, 6)\) does not belong to the graph of the function \( y = (x+1)^3 - 2 \).

Answer 2

Use the table to answer the question.

x
y

−5
8
−7
14
Determine the rate of change and the initial value of the linear function given here as a table of values.

(1 point)
Responses

m=−0.5, b=10.5
m equals negative 0.5 comma b equals 10.5

m=−0.5, b=5.5
m equals negative 0.5 comma b equals 5.5

m=−3, b=−7
m equals negative 3 comma b equals negative 7

m=−7, b=−3

Answer 3

To determine the rate of change (slope) and the initial value (y-intercept) of the linear function, we can use the two points given in the table:

1. \( (-5, 8) \)
2. \( (-7, 14) \)

Step 1: Calculate the Rate of Change (Slope)

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using our points:

• Let \( (x_1, y_1) = (-5, 8) \)
• Let \( (x_2, y_2) = (-7, 14) \)

Plugging in the values:

\[ m = \frac{14 - 8}{-7 - (-5)} = \frac{6}{-2} = -3 \]

Step 2: Determine the Initial Value (y-intercept)

To find the y-intercept \( b \), we can use the slope-intercept form \( y = mx + b \). We can use one of the points and the slope we just calculated.

Using the point \( (-5, 8) \):

\[ 8 = -3(-5) + b \] \[ 8 = 15 + b \] \[ b = 8 - 15 = -7 \]

Conclusion

The rate of change (slope) is \( m = -3 \) and the initial value (y-intercept) is \( b = -7 \). Therefore, the answer is:

\( m = -3, b = -7 \).

Answer 4

Use the table to answer the question.

Time spent driving (hours) Distance traveled (miles)
0 0
12
30
112
90
2 120
4 240
The table shows the distance traveled on a bus trip from Houston, TX to Dallas, TX as a function of time. Find the rate at which the bus is traveling.

Answer 5

To find the rate at which the bus is traveling, we can use the concept of speed, which is defined as the distance traveled divided by the time spent.

From the table, we can extract the relevant data points for calculating the average rate between the times given:

1. At 0 hours, the distance traveled is 0 miles.
2. At 2 hours, the distance traveled is 120 miles.
3. At 4 hours, the distance traveled is 240 miles.

Calculation of Speed

We can calculate the average speed over these intervals:

1. Between 0 hours and 2 hours:

• Time: \( 2 - 0 = 2 \) hours
• Distance: \( 120 - 0 = 120 \) miles

Speed (rate of travel) for this interval:

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{120 \text{ miles}}{2 \text{ hours}} = 60 \text{ miles per hour} \]

2. Between 2 hours and 4 hours:

• Time: \( 4 - 2 = 2 \) hours
• Distance: \( 240 - 120 = 120 \) miles

Speed for this interval:

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{120 \text{ miles}}{2 \text{ hours}} = 60 \text{ miles per hour} \]

Summary

From both intervals calculated, the bus is traveling at a consistent rate of 60 miles per hour.

Answer 6

Which two points on a linear graph showing time and distance would indicate a rate of change of 80 km per hour?(1 point)
Responses

(4,480)
and (6,600)
left parenthesis 4 comma 480 right parenthesis and left parenthesis 6 comma 600 right parenthesis

(5,400)
and (6,560)
left parenthesis 5 comma 400 right parenthesis and left parenthesis 6 comma 560 right parenthesis

(4,400)
and (6,480)
left parenthesis 4 comma 400 right parenthesis and left parenthesis 6 comma 480 right parenthesis

(4,320)
and (6,480)

Answer 7

To determine which two points indicate a rate of change of 80 km per hour, we can use the formula for the slope (or rate of change) between two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[ \text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} \]

We want this to equal 80 km/h.

Let's evaluate each pair of points:

1. Points \( (4, 480) \) and \( (6, 600) \)

\[ \text{Rate} = \frac{600 - 480}{6 - 4} = \frac{120}{2} = 60 \text{ km/h} \]

2. Points \( (5, 400) \) and \( (6, 560) \)

\[ \text{Rate} = \frac{560 - 400}{6 - 5} = \frac{160}{1} = 160 \text{ km/h} \]

3. Points \( (4, 400) \) and \( (6, 480) \)

\[ \text{Rate} = \frac{480 - 400}{6 - 4} = \frac{80}{2} = 40 \text{ km/h} \]

4. Points \( (4, 320) \) and \( (6, 480) \)

\[ \text{Rate} = \frac{480 - 320}{6 - 4} = \frac{160}{2} = 80 \text{ km/h} \]

Conclusion

The correct pair of points that indicates a rate of change of 80 km per hour is:

\( (4, 320) \) and \( (6, 480) \).

Answer 8

In a linear function that describes a real-world situation, what part of the function corresponds to the initial value?(1 point)
Responses

the x
-intercept
the x -intercept

the x
-value when y=1

the x -value when y equals 1

the y
-intercept
the y -intercept

the y
-value when x=1

Answer 9

In a linear function that describes a real-world situation, the part of the function that corresponds to the initial value is the y-intercept.

The y-intercept is the point where the graph of the function crosses the y-axis, which occurs when \( x = 0 \). It represents the output (or value of \( y \)) when no independent variable change has occurred (i.e., at the start or initial condition).

Therefore, the correct answer is:

the y-intercept.

Answer 10

alt="An illustration shows a coordinate plane with four quadrants. The x-axis ranges from negative 50 to 50 in ten unit increments. The y-axis ranges from negative 20 to 120 in ten unit increments. An upward slanting line is graphed on the coordinate plane. The line passes through two plotted points. The points have coordinates at left parenthesis 10 comma 70 right parenthesis and left parenthesis 25 comma 100 right parenthesis."
Write the equation of the linear function shown on the graph.

(1 point)

Answer 11

To write the equation of the linear function shown on the graph, we first need to determine the slope (rate of change) and the y-intercept of the line.

Step 1: Calculate the Slope (m)

Using the two points provided:

• Point 1: \( (10, 70) \)
• Point 2: \( (25, 100) \)

The formula for calculating the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates:

\[ m = \frac{100 - 70}{25 - 10} = \frac{30}{15} = 2 \]

Step 2: Use the Slope-Intercept Form

The slope-intercept form of a linear equation is given by:

\[ y = mx + b \]

Where:

• \( m \) is the slope
• \( b \) is the y-intercept

We already found \( m = 2 \). Now we need to find \( b \), the y-intercept. We can use one of the points to do this. Let's use the point \( (10, 70) \).

Step 3: Solve for the y-Intercept (b)

Substituting the slope and the coordinates of one point into the equation:

\[ 70 = 2(10) + b \]

Calculating:

\[ 70 = 20 + b \]

Now, solve for \( b \):

\[ b = 70 - 20 = 50 \]

Step 4: Write the Final Equation

Now that we have both the slope and the y-intercept, we can write the equation of the linear function:

\[ y = 2x + 50 \]

Conclusion

The equation of the linear function shown on the graph is:

\( y = 2x + 50 \).

Answer 12

Use the image to answer the question.

Linear Function 1: A coordinate plane shows the x-axis ranging from negative 1 to 4 in increments of 1 and the y-axis ranging from negative 1 to 13 in increments of 1. A line with arrows at both ends joins two plotted points. The coordinates of the plotted points are as follows: left parenthesis 1 comma 9 right parenthesis and left parenthesis 2 comma 12 right parenthesis. Linear Function 2:
x
y

0 2
5 42
Compare Linear Function 1 with Linear Function 2. Which one has the greater rate of change?

Answer 13

To compare the rates of change of Linear Function 1 and Linear Function 2, we need to calculate the slope (rate of change) for each linear function.

Linear Function 1

Using the two points:

• Point 1: \( (1, 9) \)
• Point 2: \( (2, 12) \)

The formula for calculating the slope \( m \) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates:

\[ m = \frac{12 - 9}{2 - 1} = \frac{3}{1} = 3 \]

Linear Function 2

Using the two points:

• Point 1: \( (0, 2) \)
• Point 2: \( (5, 42) \)

Again, we use the slope formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates:

\[ m = \frac{42 - 2}{5 - 0} = \frac{40}{5} = 8 \]

Comparison

• Slope of Linear Function 1: \( m_1 = 3 \)
• Slope of Linear Function 2: \( m_2 = 8 \)

Conclusion

Linear Function 2 has a greater rate of change, as its slope (8) is greater than that of Linear Function 1 (3).

Linear Function 2 has the greater rate of change.