Answer :
To graph the equation [tex]\( p(x) = -x^3 \)[/tex] by plotting points, follow these steps:
### Step-by-Step Solution
1. Choose a set of x-values:
Select a range of x-values that gives you a broad view of the behavior of the function. Let's choose integers from [tex]\( -3 \)[/tex] to [tex]\( 3 \)[/tex].
2. Calculate corresponding y-values:
For each x-value, compute the corresponding y-value using the function [tex]\( p(x) = -x^3 \)[/tex].
3. Plot the points:
Plot each (x, y) pair on a coordinate plane.
4. Draw the curve:
After plotting the points, connect them smoothly to show the overall shape of the function.
### Choosing x-values and calculating y-values
Let's choose the x-values [tex]\( -3, -2, -1, 0, 1, 2, 3 \)[/tex]:
[tex]\[ \begin{align*} p(-3) &= -(-3)^3 = -(-27) = 27 \\ p(-2) &= -(-2)^3 = -(-8) = 8 \\ p(-1) &= -(-1)^3 = -(-1) = 1 \\ p(0) &= -(0)^3 = 0 \\ p(1) &= -(1)^3 = -1 \\ p(2) &= -(2)^3 = -8 \\ p(3) &= -(3)^3 = -27 \\ \end{align*} \][/tex]
So, the (x, y) pairs are:
[tex]\[ \begin{align*} (-3, 27) \\ (-2, 8) \\ (-1, 1) \\ (0, 0) \\ (1, -1) \\ (2, -8) \\ (3, -27) \\ \end{align*} \][/tex]
### Plotting the Points
1. Create a coordinate plane with the x and y axes.
2. Plot each of the points (x, y) determined above.
### Drawing the Curve
Connect the plotted points with a smooth curve, as this is a continuous function. The function [tex]\( p(x) = -x^3 \)[/tex] is cubic and has the following characteristics:
- It passes through the origin (0, 0).
- For [tex]\( x > 0 \)[/tex], [tex]\( p(x) \)[/tex] is decreasing because [tex]\( -x^3 \)[/tex] is negative and becomes more negative as [tex]\( x \)[/tex] increases.
- For [tex]\( x < 0 \)[/tex], [tex]\( p(x) \)[/tex] is increasing because [tex]\( -x^3 \)[/tex] turns negative [tex]\( x \)[/tex] values into positive.
### Graph Description
- The graph will be symmetrical with respect to the origin (it is an odd function).
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( p(x) \)[/tex] will approach negative infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( p(x) \)[/tex] will approach positive infinity.
This is the shape and key properties of the graph of [tex]\( p(x) = -x^3 \)[/tex].
### Step-by-Step Solution
1. Choose a set of x-values:
Select a range of x-values that gives you a broad view of the behavior of the function. Let's choose integers from [tex]\( -3 \)[/tex] to [tex]\( 3 \)[/tex].
2. Calculate corresponding y-values:
For each x-value, compute the corresponding y-value using the function [tex]\( p(x) = -x^3 \)[/tex].
3. Plot the points:
Plot each (x, y) pair on a coordinate plane.
4. Draw the curve:
After plotting the points, connect them smoothly to show the overall shape of the function.
### Choosing x-values and calculating y-values
Let's choose the x-values [tex]\( -3, -2, -1, 0, 1, 2, 3 \)[/tex]:
[tex]\[ \begin{align*} p(-3) &= -(-3)^3 = -(-27) = 27 \\ p(-2) &= -(-2)^3 = -(-8) = 8 \\ p(-1) &= -(-1)^3 = -(-1) = 1 \\ p(0) &= -(0)^3 = 0 \\ p(1) &= -(1)^3 = -1 \\ p(2) &= -(2)^3 = -8 \\ p(3) &= -(3)^3 = -27 \\ \end{align*} \][/tex]
So, the (x, y) pairs are:
[tex]\[ \begin{align*} (-3, 27) \\ (-2, 8) \\ (-1, 1) \\ (0, 0) \\ (1, -1) \\ (2, -8) \\ (3, -27) \\ \end{align*} \][/tex]
### Plotting the Points
1. Create a coordinate plane with the x and y axes.
2. Plot each of the points (x, y) determined above.
### Drawing the Curve
Connect the plotted points with a smooth curve, as this is a continuous function. The function [tex]\( p(x) = -x^3 \)[/tex] is cubic and has the following characteristics:
- It passes through the origin (0, 0).
- For [tex]\( x > 0 \)[/tex], [tex]\( p(x) \)[/tex] is decreasing because [tex]\( -x^3 \)[/tex] is negative and becomes more negative as [tex]\( x \)[/tex] increases.
- For [tex]\( x < 0 \)[/tex], [tex]\( p(x) \)[/tex] is increasing because [tex]\( -x^3 \)[/tex] turns negative [tex]\( x \)[/tex] values into positive.
### Graph Description
- The graph will be symmetrical with respect to the origin (it is an odd function).
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( p(x) \)[/tex] will approach negative infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( p(x) \)[/tex] will approach positive infinity.
This is the shape and key properties of the graph of [tex]\( p(x) = -x^3 \)[/tex].
