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is the graph linear, nonlinear, function or not a function

x y
1 1
2 8
3 27
4 64
5 125
6 216
7 ?

11 answers

To determine whether the given graph is linear, nonlinear, a function, or not a function, let's take a look at the provided values for \(x\) and \(y\).

The values are:

  • \(x = 1\), \(y = 1\)
  • \(x = 2\), \(y = 8\)
  • \(x = 3\), \(y = 27\)
  • \(x = 4\), \(y = 64\)
  • \(x = 5\), \(y = 125\)
  • \(x = 6\), \(y = 216\)

From the pattern of the \(y\) values, they can be expressed as powers of \(x\):

  • \(y(1) = 1^3 = 1\)
  • \(y(2) = 2^3 = 8\)
  • \(y(3) = 3^3 = 27\)
  • \(y(4) = 4^3 = 64\)
  • \(y(5) = 5^3 = 125\)
  • \(y(6) = 6^3 = 216\)

This suggests that \(y = x^3\).

For \(x = 7\):

  • \(y(7) = 7^3 = 343\)

Analysis:

  1. Linear vs. Nonlinear: Since \(y\) does not form a straight line when plotted (as it involves cubic powers), this is a nonlinear relationship.
  2. Function vs. Not a Function: Each \(x\) value corresponds to exactly one \(y\) value. Therefore, it meets the definition of a function.

Conclusion:

The graph represented by the given points is nonlinear and is a function.

is (y+7)=3(x−2) linear, nonlinear, function, or not a function

To analyze the equation \((y + 7) = 3(x - 2)\), we can rewrite it in a more standard form to determine its characteristics.

  1. Rearranging the Equation: \[ y + 7 = 3(x - 2) \] \[ y + 7 = 3x - 6 \] \[ y = 3x - 6 - 7 \] \[ y = 3x - 13 \]

  2. Identifying Linear vs. Nonlinear: The equation \(y = 3x - 13\) is in the slope-intercept form \(y = mx + b\) where \(m\) is the slope (3 in this case) and \(b\) is the y-intercept (-13 in this case). This indicates that the equation represents a straight line, which means it is linear.

  3. Identifying Function vs. Not a Function: In this equation, for every value of \(x\), there is exactly one corresponding value of \(y\). Therefore, it meets the definition of a function.

Conclusion:

The equation \((y + 7) = 3(x - 2)\) is linear and it is a function.

is the graph linear, nonlinear, function, or not a function
time (hours temperature (degree Celsius)
3 5
6 0
9 -5
12 -10

To determine whether the relationship between time (in hours) and temperature (in degrees Celsius) is linear, nonlinear, a function, or not a function, let's analyze the provided data:

Data Points:

  • Time (hours): 3, 6, 9, 12
  • Temperature (°C): 5, 0, -5, -10

These points are:

  1. (3, 5)
  2. (6, 0)
  3. (9, -5)
  4. (12, -10)

Analysis:

  1. Function vs. Not a Function:

    • To determine if it's a function, we check if each \(x\) value corresponds to exactly one \(y\) value. In this case, for each hour (3, 6, 9, 12) there is a unique temperature. Therefore, this relationship is a function.

  2. Linear vs. Nonlinear:

    • We can check the differences in the \(y\) values as \(x\) increases to see if the relationship appears to be linear:
      • From 3 to 6: \(0 - 5 = -5\)
      • From 6 to 9: \(-5 - 0 = -5\)
      • From 9 to 12: \(-10 - (-5) = -5\)

    The change in temperature is consistent (a decrease of 5 degrees Celsius for each increase of 3 hours). This indicates a constant rate of change.

Conclusion:

Since the relationship is a function and exhibits a constant rate of change (indicating a straight line when graphed), the graph of the data points is linear and it is a function.

does the graph show a function?
x y
1 -8
2 -6
3 -1
4 -2
5 -4

To determine whether the given set of points (x, y) represents a function, we need to check if each \(x\) value corresponds to exactly one unique \(y\) value.

Given Points:

  • \(x = 1\), \(y = -8\)
  • \(x = 2\), \(y = -6\)
  • \(x = 3\), \(y = -1\)
  • \(x = 4\), \(y = -2\)
  • \(x = 5\), \(y = -4\)

Check for Function:

Let's analyze the unique \(y\) values for the given \(x\) values:

  • For \(x = 1\) → \(y = -8\)
  • For \(x = 2\) → \(y = -6\)
  • For \(x = 3\) → \(y = -1\)
  • For \(x = 4\) → \(y = -2\)
  • For \(x = 5\) → \(y = -4\)

Conclusion:

Each \(x\) value corresponds to a unique \(y\) value, meaning no \(x\) value is repeated with a different \(y\) value.

Thus, yes, the graph represented by these points shows a function.

is this a function?{(0, 20), (2, 70), (10, 45), (1, 30), (3, 70)}

To determine whether the given set of points represents a function, we need to check if each \(x\) value corresponds to exactly one unique \(y\) value.

Given Points:

  • \((0, 20)\)
  • \((2, 70)\)
  • \((10, 45)\)
  • \((1, 30)\)
  • \((3, 70)\)

Check for Function:

Now let's look at the \(x\)-values:

  • \(0\) → \(y = 20\)
  • \(2\) → \(y = 70\)
  • \(10\) → \(y = 45\)
  • \(1\) → \(y = 30\)
  • \(3\) → \(y = 70\)

Notice that \(x = 2\) and \(x = 3\) both correspond to \(y = 70\), since \(y = 70\) appears twice for different \(x\) values.

Conclusion:

Since the \(x\) value \(2\) and \(3\) both lead to the \(y\) value \(70\), this means that there is not a unique \(y\) value for every \(x\). Hence, the given set of points is not a function.

Look at the graph and answer the following questions:
the graph starts at 0, 0 then goes up to (10, 60) then moves a little to the right and is now at (15, 60) then goes down to (40, -40) and goes back up to (55, 0)

(5 points)
1) What interval shows a constant velocity over time? (Response area)

2) What interval shows the graph decreasing? (Response area)

3) After 20 minutes , what would the velocity be according to the graph? (Response area)

4) What is the domain for this function? (Response area)

5) Is this linear or nonlinear? (Response area)

responses
nonlinear
(0,55)
(10,15)
60 m/min
(14,40)
40 m/min
linear
(-40,60)
(0,10)

Based on your description of the graph, I will answer the questions according to the provided options for responses.

1) What interval shows a constant velocity over time?

From the description, the graph is flat between the points \((10, 60)\) and \((15, 60)\). This indicates that the velocity is constant (and zero) during that time period. Response: (10, 15)

2) What interval shows the graph decreasing?

The graph decreases between the points \((15, 60)\) and \((40, -40)\). This means that between these two points, the value of \(y\) is getting lower. Response: (15, 40)

3) After 20 minutes, what would the velocity be according to the graph?

At 20 minutes, according to your description, the graph has dropped from \((15, 60)\) to \((40, -40)\). The graph indicates a downward trend during that section. To be more specific about the velocity:

  • The point at \(20\) minutes is between \((15, 60)\) and \((40, -40)\).
  • The specific calculation would be a decrease in \(y\) from \(60\) to \(-40\) over \(25\) minutes: \[ \text{velocity} = \frac{-40 - 60}{40 - 15} = \frac{-100}{25} = -4 \text{ units/min} \] So, the immediate velocity can be interpreted through the slope (although this context may suggest a more generic velocity reading across the graph).

Response: (assuming it's expected to be interpreted around this general context): -4 units/min.

4) What is the domain for this function?

The function starts at \(x = 0\) and ends at \(x = 55\). Therefore, the domain in interval notation is from 0 to 55. Response: (0, 55)

5) Is this linear or nonlinear?

Since the graph has flat sections and slopes that change (increasing and decreasing segments, not forming a straight line), the graph is nonlinear. Response: nonlinear

Summary of Responses:

  1. (10, 15)
  2. (15, 40)
  3. -4 units/min (interpretively)
  4. (0, 55)
  5. nonlinear