Pre-Algebra Examples

$h (x) = {(7 - 5 x)}^{3} - 6 {(7 - 5 x)}^{2} + 4 (7 - 5 x) - 1$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

$f (- 1) = {(7 - 5 \cdot - 1)}^{3} - 6 {(7 - 5 \cdot - 1)}^{2} - 20 \cdot - 1 + 27$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $- 5$ by $- 1$ .

$f (- 1) = {(7 + 5)}^{3} - 6 {(7 - 5 \cdot - 1)}^{2} - 20 \cdot - 1 + 27$

Step 1.2.1.2

Add $7$ and $5$ .

$f (- 1) = 12^{3} - 6 {(7 - 5 \cdot - 1)}^{2} - 20 \cdot - 1 + 27$

Step 1.2.1.3

Raise $12$ to the power of $3$ .

$f (- 1) = 1728 - 6 {(7 - 5 \cdot - 1)}^{2} - 20 \cdot - 1 + 27$

Step 1.2.1.4

Multiply $- 5$ by $- 1$ .

$f (- 1) = 1728 - 6 {(7 + 5)}^{2} - 20 \cdot - 1 + 27$

Step 1.2.1.5

Add $7$ and $5$ .

$f (- 1) = 1728 - 6 \cdot 12^{2} - 20 \cdot - 1 + 27$

Step 1.2.1.6

Raise $12$ to the power of $2$ .

$f (- 1) = 1728 - 6 \cdot 144 - 20 \cdot - 1 + 27$

Step 1.2.1.7

Multiply $- 6$ by $144$ .

$f (- 1) = 1728 - 864 - 20 \cdot - 1 + 27$

Step 1.2.1.8

Multiply $- 20$ by $- 1$ .

$f (- 1) = 1728 - 864 + 20 + 27$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Subtract $864$ from $1728$ .

$f (- 1) = 864 + 20 + 27$

Step 1.2.2.2

Add $864$ and $20$ .

$f (- 1) = 884 + 27$

Step 1.2.2.3

Add $884$ and $27$ .

$f (- 1) = 911$

Step 1.2.3

The final answer is $911$ .

$911$

Step 1.3

Convert $911$ to decimal.

$y = 911$

Step 2

Find the point at $x = 0$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(7 - 5 \cdot 0)}^{3} - 6 {(7 - 5 \cdot 0)}^{2} - 20 \cdot 0 + 27$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $- 5$ by $0$ .

$f (0) = {(7 + 0)}^{3} - 6 {(7 - 5 \cdot 0)}^{2} - 20 \cdot 0 + 27$

Step 2.2.1.2

Add $7$ and $0$ .

$f (0) = 7^{3} - 6 {(7 - 5 \cdot 0)}^{2} - 20 \cdot 0 + 27$

Step 2.2.1.3

Raise $7$ to the power of $3$ .

$f (0) = 343 - 6 {(7 - 5 \cdot 0)}^{2} - 20 \cdot 0 + 27$

Step 2.2.1.4

Multiply $- 5$ by $0$ .

$f (0) = 343 - 6 {(7 + 0)}^{2} - 20 \cdot 0 + 27$

Step 2.2.1.5

Add $7$ and $0$ .

$f (0) = 343 - 6 \cdot 7^{2} - 20 \cdot 0 + 27$

Step 2.2.1.6

Raise $7$ to the power of $2$ .

$f (0) = 343 - 6 \cdot 49 - 20 \cdot 0 + 27$

Step 2.2.1.7

Multiply $- 6$ by $49$ .

$f (0) = 343 - 294 - 20 \cdot 0 + 27$

Step 2.2.1.8

Multiply $- 20$ by $0$ .

$f (0) = 343 - 294 + 0 + 27$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $294$ from $343$ .

$f (0) = 49 + 0 + 27$

Step 2.2.2.2

Add $49$ and $0$ .

$f (0) = 49 + 27$

Step 2.2.2.3

Add $49$ and $27$ .

$f (0) = 76$

Step 2.2.3

The final answer is $76$ .

$76$

Step 2.3

Convert $76$ to decimal.

$y = 76$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

$f (1) = {(7 - 5 \cdot 1)}^{3} - 6 {(7 - 5 \cdot 1)}^{2} - 20 \cdot 1 + 27$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $- 5$ by $1$ .

$f (1) = {(7 - 5)}^{3} - 6 {(7 - 5 \cdot 1)}^{2} - 20 \cdot 1 + 27$

Step 3.2.1.2

Subtract $5$ from $7$ .

$f (1) = 2^{3} - 6 {(7 - 5 \cdot 1)}^{2} - 20 \cdot 1 + 27$

Step 3.2.1.3

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

$f (1) = 8 - 6 {(7 - 5 \cdot 1)}^{2} - 20 \cdot 1 + 27$

Step 3.2.1.4

Multiply $- 5$ by $1$ .

$f (1) = 8 - 6 {(7 - 5)}^{2} - 20 \cdot 1 + 27$

Step 3.2.1.5

Subtract $5$ from $7$ .

$f (1) = 8 - 6 \cdot 2^{2} - 20 \cdot 1 + 27$

Step 3.2.1.6

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

$f (1) = 8 - 6 \cdot 4 - 20 \cdot 1 + 27$

Step 3.2.1.7

Multiply $- 6$ by $4$ .

$f (1) = 8 - 24 - 20 \cdot 1 + 27$

Step 3.2.1.8

Multiply $- 20$ by $1$ .

$f (1) = 8 - 24 - 20 + 27$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $24$ from $8$ .

$f (1) = - 16 - 20 + 27$

Step 3.2.2.2

Subtract $20$ from $- 16$ .

$f (1) = - 36 + 27$

Step 3.2.2.3

Add $- 36$ and $27$ .

$f (1) = - 9$

Step 3.2.3

The final answer is $- 9$ .

$- 9$

Step 3.3

Convert $- 9$ to decimal.

$y = - 9$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = {(7 - 5 \cdot 2)}^{3} - 6 {(7 - 5 \cdot 2)}^{2} - 20 \cdot 2 + 27$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $- 5$ by $2$ .

$f (2) = {(7 - 10)}^{3} - 6 {(7 - 5 \cdot 2)}^{2} - 20 \cdot 2 + 27$

Step 4.2.1.2

Subtract $10$ from $7$ .

$f (2) = {(- 3)}^{3} - 6 {(7 - 5 \cdot 2)}^{2} - 20 \cdot 2 + 27$

Step 4.2.1.3

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

$f (2) = - 27 - 6 {(7 - 5 \cdot 2)}^{2} - 20 \cdot 2 + 27$

Step 4.2.1.4

Multiply $- 5$ by $2$ .

$f (2) = - 27 - 6 {(7 - 10)}^{2} - 20 \cdot 2 + 27$

Step 4.2.1.5

Subtract $10$ from $7$ .

$f (2) = - 27 - 6 {(- 3)}^{2} - 20 \cdot 2 + 27$

Step 4.2.1.6

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

$f (2) = - 27 - 6 \cdot 9 - 20 \cdot 2 + 27$

Step 4.2.1.7

Multiply $- 6$ by $9$ .

$f (2) = - 27 - 54 - 20 \cdot 2 + 27$

Step 4.2.1.8

Multiply $- 20$ by $2$ .

$f (2) = - 27 - 54 - 40 + 27$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $54$ from $- 27$ .

$f (2) = - 81 - 40 + 27$

Step 4.2.2.2

Subtract $40$ from $- 81$ .

$f (2) = - 121 + 27$

Step 4.2.2.3

Add $- 121$ and $27$ .

$f (2) = - 94$

Step 4.2.3

The final answer is $- 94$ .

$- 94$

Step 4.3

Convert $- 94$ to decimal.

$y = - 94$

Step 5

Find the point at $x = 3$ .

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Step 5.1

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

$f (3) = {(7 - 5 \cdot 3)}^{3} - 6 {(7 - 5 \cdot 3)}^{2} - 20 \cdot 3 + 27$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $- 5$ by $3$ .

$f (3) = {(7 - 15)}^{3} - 6 {(7 - 5 \cdot 3)}^{2} - 20 \cdot 3 + 27$

Step 5.2.1.2

Subtract $15$ from $7$ .

$f (3) = {(- 8)}^{3} - 6 {(7 - 5 \cdot 3)}^{2} - 20 \cdot 3 + 27$

Step 5.2.1.3

Raise $- 8$ to the power of $3$ .

$f (3) = - 512 - 6 {(7 - 5 \cdot 3)}^{2} - 20 \cdot 3 + 27$

Step 5.2.1.4

Multiply $- 5$ by $3$ .

$f (3) = - 512 - 6 {(7 - 15)}^{2} - 20 \cdot 3 + 27$

Step 5.2.1.5

Subtract $15$ from $7$ .

$f (3) = - 512 - 6 {(- 8)}^{2} - 20 \cdot 3 + 27$

Step 5.2.1.6

Raise $- 8$ to the power of $2$ .

$f (3) = - 512 - 6 \cdot 64 - 20 \cdot 3 + 27$

Step 5.2.1.7

Multiply $- 6$ by $64$ .

$f (3) = - 512 - 384 - 20 \cdot 3 + 27$

Step 5.2.1.8

Multiply $- 20$ by $3$ .

$f (3) = - 512 - 384 - 60 + 27$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $384$ from $- 512$ .

$f (3) = - 896 - 60 + 27$

Step 5.2.2.2

Subtract $60$ from $- 896$ .

$f (3) = - 956 + 27$

Step 5.2.2.3

Add $- 956$ and $27$ .

$f (3) = - 929$

Step 5.2.3

The final answer is $- 929$ .

$- 929$

Step 5.3

Convert $- 929$ to decimal.

$y = - 929$

Step 6

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

$\begin{array}{c} x & y \\ - 1 & 911 \\ 0 & 76 \\ 1 & - 9 \\ 2 & - 94 \\ 3 & - 929 \end{array}$

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

$\begin{array}{c} x & y \\ - 1 & 911 \\ 0 & 76 \\ 1 & - 9 \\ 2 & - 94 \\ 3 & - 929 \end{array}$

Step 8