Calculus Examples

$f (x) = - x^{3} + 9 x^{2} - 51$

Step 1

Find the point at $x = 1$ .

Tap for more steps...

Step 1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - {(1)}^{3} + 9 {(1)}^{2} - 51$

Step 1.2

Simplify the result.

Tap for more steps...

Step 1.2.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1

One to any power is one.

$f (1) = - 1 \cdot 1 + 9 {(1)}^{2} - 51$

Step 1.2.1.2

Multiply $- 1$ by $1$ .

$f (1) = - 1 + 9 {(1)}^{2} - 51$

Step 1.2.1.3

One to any power is one.

$f (1) = - 1 + 9 \cdot 1 - 51$

Step 1.2.1.4

Multiply $9$ by $1$ .

$f (1) = - 1 + 9 - 51$

Step 1.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 1.2.2.1

Add $- 1$ and $9$ .

$f (1) = 8 - 51$

Step 1.2.2.2

Subtract $51$ from $8$ .

$f (1) = - 43$

Step 1.2.3

The final answer is $- 43$ .

$- 43$

Step 1.3

Convert $- 43$ to decimal.

$y = - 43$

Step 2

Find the point at $x = 2$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = - {(2)}^{3} + 9 {(2)}^{2} - 51$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Raise $2$ to the power of $3$ .

$f (2) = - 1 \cdot 8 + 9 {(2)}^{2} - 51$

Step 2.2.1.2

Multiply $- 1$ by $8$ .

$f (2) = - 8 + 9 {(2)}^{2} - 51$

Step 2.2.1.3

Raise $2$ to the power of $2$ .

$f (2) = - 8 + 9 \cdot 4 - 51$

Step 2.2.1.4

Multiply $9$ by $4$ .

$f (2) = - 8 + 36 - 51$

Step 2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.2.1

Add $- 8$ and $36$ .

$f (2) = 28 - 51$

Step 2.2.2.2

Subtract $51$ from $28$ .

$f (2) = - 23$

Step 2.2.3

The final answer is $- 23$ .

$- 23$

Step 2.3

Convert $- 23$ to decimal.

$y = - 23$

Step 3

Find the point at $x = 3$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = - {(3)}^{3} + 9 {(3)}^{2} - 51$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Raise $3$ to the power of $3$ .

$f (3) = - 1 \cdot 27 + 9 {(3)}^{2} - 51$

Step 3.2.1.2

Multiply $- 1$ by $27$ .

$f (3) = - 27 + 9 {(3)}^{2} - 51$

Step 3.2.1.3

Raise $3$ to the power of $2$ .

$f (3) = - 27 + 9 \cdot 9 - 51$

Step 3.2.1.4

Multiply $9$ by $9$ .

$f (3) = - 27 + 81 - 51$

Step 3.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 3.2.2.1

Add $- 27$ and $81$ .

$f (3) = 54 - 51$

Step 3.2.2.2

Subtract $51$ from $54$ .

$f (3) = 3$

Step 3.2.3

The final answer is $3$ .

$3$

Step 3.3

Convert $3$ to decimal.

$y = 3$

Step 4

Find the point at $x = 4$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = - {(4)}^{3} + 9 {(4)}^{2} - 51$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Raise $4$ to the power of $3$ .

$f (4) = - 1 \cdot 64 + 9 {(4)}^{2} - 51$

Step 4.2.1.2

Multiply $- 1$ by $64$ .

$f (4) = - 64 + 9 {(4)}^{2} - 51$

Step 4.2.1.3

Raise $4$ to the power of $2$ .

$f (4) = - 64 + 9 \cdot 16 - 51$

Step 4.2.1.4

Multiply $9$ by $16$ .

$f (4) = - 64 + 144 - 51$

Step 4.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.2.1

Add $- 64$ and $144$ .

$f (4) = 80 - 51$

Step 4.2.2.2

Subtract $51$ from $80$ .

$f (4) = 29$

Step 4.2.3

The final answer is $29$ .

$29$

Step 4.3

Convert $29$ to decimal.

$y = 29$

Step 5

Find the point at $x = 5$ .

Tap for more steps...

Step 5.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - {(5)}^{3} + 9 {(5)}^{2} - 51$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Raise $5$ to the power of $3$ .

$f (5) = - 1 \cdot 125 + 9 {(5)}^{2} - 51$

Step 5.2.1.2

Multiply $- 1$ by $125$ .

$f (5) = - 125 + 9 {(5)}^{2} - 51$

Step 5.2.1.3

Raise $5$ to the power of $2$ .

$f (5) = - 125 + 9 \cdot 25 - 51$

Step 5.2.1.4

Multiply $9$ by $25$ .

$f (5) = - 125 + 225 - 51$

Step 5.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 5.2.2.1

Add $- 125$ and $225$ .

$f (5) = 100 - 51$

Step 5.2.2.2

Subtract $51$ from $100$ .

$f (5) = 49$

Step 5.2.3

The final answer is $49$ .

$49$

Step 5.3

Convert $49$ to decimal.

$y = 49$

Step 6

The cubic function can be graphed using the function behavior and the points.

$\begin{array}{c} x & y \\ 1 & - 43 \\ 2 & - 23 \\ 3 & 3 \\ 4 & 29 \\ 5 & 49 \end{array}$

Step 7

The cubic function can be graphed using the function behavior and the selected points.

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ 1 & - 43 \\ 2 & - 23 \\ 3 & 3 \\ 4 & 29 \\ 5 & 49 \end{array}$

Step 8