Question
Find seven ordered pairs for the equation y equals x cubed plus 7 using the given values of x. Then determine its graph
Answers
GPT-4o mini
To find seven ordered pairs for the equation \( y = x^3 + 7 \), we can use chosen values for \( x \). Let's choose the values of \( x \) from -3 to 3 (inclusive), providing a symmetrical range around 0.
Here are the calculations for the ordered pairs:
1. For \( x = -3 \):
\[
y = (-3)^3 + 7 = -27 + 7 = -20 \quad \Rightarrow \quad (-3, -20)
\]
2. For \( x = -2 \):
\[
y = (-2)^3 + 7 = -8 + 7 = -1 \quad \Rightarrow \quad (-2, -1)
\]
3. For \( x = -1 \):
\[
y = (-1)^3 + 7 = -1 + 7 = 6 \quad \Rightarrow \quad (-1, 6)
\]
4. For \( x = 0 \):
\[
y = 0^3 + 7 = 0 + 7 = 7 \quad \Rightarrow \quad (0, 7)
\]
5. For \( x = 1 \):
\[
y = 1^3 + 7 = 1 + 7 = 8 \quad \Rightarrow \quad (1, 8)
\]
6. For \( x = 2 \):
\[
y = 2^3 + 7 = 8 + 7 = 15 \quad \Rightarrow \quad (2, 15)
\]
7. For \( x = 3 \):
\[
y = 3^3 + 7 = 27 + 7 = 34 \quad \Rightarrow \quad (3, 34)
\]
Now we can list the seven ordered pairs:
1. \( (-3, -20) \)
2. \( (-2, -1) \)
3. \( (-1, 6) \)
4. \( (0, 7) \)
5. \( (1, 8) \)
6. \( (2, 15) \)
7. \( (3, 34) \)
### Graphing the Equation
To graph the equation \( y = x^3 + 7 \), we can plot these points on a Cartesian coordinate system:
- The point \( (-3, -20) \) is far down on the y-axis.
- The point \( (-2, -1) \) is just below the x-axis.
- The point \( (-1, 6) \) is in the first quadrant.
- The point \( (0, 7) \) is above the origin.
- The point \( (1, 8) \) is also in the first quadrant.
- The point \( (2, 15) \) continues going up.
- The point \( (3, 34) \) is significantly higher.
The graph of \( y = x^3 + 7 \) is characterized by:
- A cubic shape that passes through the calculated points.
- It decreases to the left of the origin and increases significantly to the right.
- The overall shape is symmetrical about the origin when shifted up by 7 units.
To visualize it, you would typically use graphing software or manual graph paper to get a more accurate representation of the curve. The curve will rise steeply for positive \( x \) and descend for negative \( x \).
Here are the calculations for the ordered pairs:
1. For \( x = -3 \):
\[
y = (-3)^3 + 7 = -27 + 7 = -20 \quad \Rightarrow \quad (-3, -20)
\]
2. For \( x = -2 \):
\[
y = (-2)^3 + 7 = -8 + 7 = -1 \quad \Rightarrow \quad (-2, -1)
\]
3. For \( x = -1 \):
\[
y = (-1)^3 + 7 = -1 + 7 = 6 \quad \Rightarrow \quad (-1, 6)
\]
4. For \( x = 0 \):
\[
y = 0^3 + 7 = 0 + 7 = 7 \quad \Rightarrow \quad (0, 7)
\]
5. For \( x = 1 \):
\[
y = 1^3 + 7 = 1 + 7 = 8 \quad \Rightarrow \quad (1, 8)
\]
6. For \( x = 2 \):
\[
y = 2^3 + 7 = 8 + 7 = 15 \quad \Rightarrow \quad (2, 15)
\]
7. For \( x = 3 \):
\[
y = 3^3 + 7 = 27 + 7 = 34 \quad \Rightarrow \quad (3, 34)
\]
Now we can list the seven ordered pairs:
1. \( (-3, -20) \)
2. \( (-2, -1) \)
3. \( (-1, 6) \)
4. \( (0, 7) \)
5. \( (1, 8) \)
6. \( (2, 15) \)
7. \( (3, 34) \)
### Graphing the Equation
To graph the equation \( y = x^3 + 7 \), we can plot these points on a Cartesian coordinate system:
- The point \( (-3, -20) \) is far down on the y-axis.
- The point \( (-2, -1) \) is just below the x-axis.
- The point \( (-1, 6) \) is in the first quadrant.
- The point \( (0, 7) \) is above the origin.
- The point \( (1, 8) \) is also in the first quadrant.
- The point \( (2, 15) \) continues going up.
- The point \( (3, 34) \) is significantly higher.
The graph of \( y = x^3 + 7 \) is characterized by:
- A cubic shape that passes through the calculated points.
- It decreases to the left of the origin and increases significantly to the right.
- The overall shape is symmetrical about the origin when shifted up by 7 units.
To visualize it, you would typically use graphing software or manual graph paper to get a more accurate representation of the curve. The curve will rise steeply for positive \( x \) and descend for negative \( x \).