Precalculus Examples

$- {(t - 1)}^{2} + 5$

Step 1

Write $- {(t - 1)}^{2} + 5$ as a function.

$f (t) = - {(t - 1)}^{2} + 5$

Step 2

Consider the difference quotient formula.

$\frac{f (t + h) - f t}{h}$

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = t + h$ .

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Step 3.1.1

Replace the variable $t$ with $t + h$ in the expression.

$f (t + h) = - {((t + h) - 1)}^{2} + 5$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Rewrite ${(t + h - 1)}^{2}$ as $(t + h - 1) (t + h - 1)$ .

$f (t + h) = - ((t + h - 1) (t + h - 1)) + 5$

Step 3.1.2.1.2

Expand $(t + h - 1) (t + h - 1)$ by multiplying each term in the first expression by each term in the second expression.

$f (t + h) = - (t \cdot t + t h + t \cdot - 1 + h t + h \cdot h + h \cdot - 1 - 1 t - 1 h - 1 \cdot - 1) + 5$

Step 3.1.2.1.3

Simplify each term.

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Step 3.1.2.1.3.1

Multiply $t$ by $t$ .

$f (t + h) = - (t^{2} + t h + t \cdot - 1 + h t + h \cdot h + h \cdot - 1 - 1 t - 1 h - 1 \cdot - 1) + 5$

Step 3.1.2.1.3.2

Move $- 1$ to the left of $t$ .

$f (t + h) = - (t^{2} + t h - 1 \cdot t + h t + h \cdot h + h \cdot - 1 - 1 t - 1 h - 1 \cdot - 1) + 5$

Step 3.1.2.1.3.3

Rewrite $- 1 t$ as $- t$ .

$f (t + h) = - (t^{2} + t h - t + h t + h \cdot h + h \cdot - 1 - 1 t - 1 h - 1 \cdot - 1) + 5$

Step 3.1.2.1.3.4

Multiply $h$ by $h$ .

$f (t + h) = - (t^{2} + t h - t + h t + h^{2} + h \cdot - 1 - 1 t - 1 h - 1 \cdot - 1) + 5$

Step 3.1.2.1.3.5

Move $- 1$ to the left of $h$ .

$f (t + h) = - (t^{2} + t h - t + h t + h^{2} - 1 \cdot h - 1 t - 1 h - 1 \cdot - 1) + 5$

Step 3.1.2.1.3.6

Rewrite $- 1 h$ as $- h$ .

$f (t + h) = - (t^{2} + t h - t + h t + h^{2} - h - 1 t - 1 h - 1 \cdot - 1) + 5$

Step 3.1.2.1.3.7

Rewrite $- 1 t$ as $- t$ .

$f (t + h) = - (t^{2} + t h - t + h t + h^{2} - h - t - 1 h - 1 \cdot - 1) + 5$

Step 3.1.2.1.3.8

Rewrite $- 1 h$ as $- h$ .

$f (t + h) = - (t^{2} + t h - t + h t + h^{2} - h - t - h - 1 \cdot - 1) + 5$

Step 3.1.2.1.3.9

Multiply $- 1$ by $- 1$ .

$f (t + h) = - (t^{2} + t h - t + h t + h^{2} - h - t - h + 1) + 5$

Step 3.1.2.1.4

Add $t h$ and $h t$ .

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Step 3.1.2.1.4.1

Reorder $t$ and $h$ .

$f (t + h) = - (t^{2} + h t + h t - t + h^{2} - h - t - h + 1) + 5$

Step 3.1.2.1.4.2

Add $h t$ and $h t$ .

$f (t + h) = - (t^{2} + 2 h t - t + h^{2} - h - t - h + 1) + 5$

Step 3.1.2.1.5

Subtract $t$ from $- t$ .

$f (t + h) = - (t^{2} + 2 h t + h^{2} - h - 2 t - h + 1) + 5$

Step 3.1.2.1.6

Subtract $h$ from $- h$ .

$f (t + h) = - (t^{2} + 2 h t + h^{2} - 2 h - 2 t + 1) + 5$

Step 3.1.2.1.7

Apply the distributive property.

$f (t + h) = - t^{2} - (2 h t) - h^{2} - (- 2 h) - (- 2 t) - 1 \cdot 1 + 5$

Step 3.1.2.1.8

Simplify.

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Step 3.1.2.1.8.1

Multiply $2$ by $- 1$ .

$f (t + h) = - t^{2} - 2 (h t) - h^{2} - (- 2 h) - (- 2 t) - 1 \cdot 1 + 5$

Step 3.1.2.1.8.2

Multiply $- 2$ by $- 1$ .

$f (t + h) = - t^{2} - 2 (h t) - h^{2} + 2 h - (- 2 t) - 1 \cdot 1 + 5$

Step 3.1.2.1.8.3

Multiply $- 2$ by $- 1$ .

$f (t + h) = - t^{2} - 2 (h t) - h^{2} + 2 h + 2 t - 1 \cdot 1 + 5$

Step 3.1.2.1.8.4

Multiply $- 1$ by $1$ .

$f (t + h) = - t^{2} - 2 (h t) - h^{2} + 2 h + 2 t - 1 + 5$

$f (t + h) = - t^{2} - 2 h t - h^{2} + 2 h + 2 t - 1 + 5$

Step 3.1.2.2

Add $- 1$ and $5$ .

$f (t + h) = - t^{2} - 2 h t - h^{2} + 2 h + 2 t + 4$

Step 3.1.2.3

The final answer is $- t^{2} - 2 h t - h^{2} + 2 h + 2 t + 4$ .

$- t^{2} - 2 h t - h^{2} + 2 h + 2 t + 4$

Step 3.2

Reorder.

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Step 3.2.1

Move $- t^{2}$ .

$- 2 h t - h^{2} - t^{2} + 2 h + 2 t + 4$

Step 3.2.2

Reorder $- 2 h t$ and $- h^{2}$ .

$- h^{2} - 2 h t - t^{2} + 2 h + 2 t + 4$

Step 3.3

Find the components of the definition.

$f (t + h) = - h^{2} - 2 h t - t^{2} + 2 h + 2 t + 4$

$f (t) = - t^{2} + 2 t + 4$

$f (t + h) = - h^{2} - 2 h t - t^{2} + 2 h + 2 t + 4$

$f (t) = - t^{2} + 2 t + 4$

Step 4

Plug in the components.

$\frac{f (t + h) - f t}{h} = \frac{- h^{2} - 2 h t - t^{2} + 2 h + 2 t + 4 - (- t^{2} + 2 t + 4)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{- h^{2} - 2 h t - t^{2} + 2 h + 2 t + 4 - - t^{2} - (2 t) - 1 \cdot 4}{h}$

Step 5.1.2

Simplify.

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Step 5.1.2.1

Multiply $- - t^{2}$ .

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Step 5.1.2.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- h^{2} - 2 h t - t^{2} + 2 h + 2 t + 4 + 1 t^{2} - (2 t) - 1 \cdot 4}{h}$

Step 5.1.2.1.2

Multiply $t^{2}$ by $1$ .

$\frac{- h^{2} - 2 h t - t^{2} + 2 h + 2 t + 4 + t^{2} - (2 t) - 1 \cdot 4}{h}$

Step 5.1.2.2

Multiply $2$ by $- 1$ .

$\frac{- h^{2} - 2 h t - t^{2} + 2 h + 2 t + 4 + t^{2} - 2 t - 1 \cdot 4}{h}$

Step 5.1.2.3

Multiply $- 1$ by $4$ .

$\frac{- h^{2} - 2 h t - t^{2} + 2 h + 2 t + 4 + t^{2} - 2 t - 4}{h}$

Step 5.1.3

Add $- t^{2}$ and $t^{2}$ .

$\frac{- h^{2} - 2 h t + 2 h + 2 t + 4 + 0 - 2 t - 4}{h}$

Step 5.1.4

Add $- h^{2}$ and $0$ .

$\frac{- h^{2} - 2 h t + 2 h + 2 t + 4 - 2 t - 4}{h}$

Step 5.1.5

Subtract $2 t$ from $2 t$ .

$\frac{- h^{2} - 2 h t + 2 h + 0 + 4 - 4}{h}$

Step 5.1.6

Add $- h^{2}$ and $0$ .

$\frac{- h^{2} - 2 h t + 2 h + 4 - 4}{h}$

Step 5.1.7

Subtract $4$ from $4$ .

$\frac{- h^{2} - 2 h t + 2 h + 0}{h}$

Step 5.1.8

Add $- h^{2} - 2 h t + 2 h$ and $0$ .

$\frac{- h^{2} - 2 h t + 2 h}{h}$

Step 5.1.9

Factor $h$ out of $- h^{2} - 2 h t + 2 h$ .

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Step 5.1.9.1

Factor $h$ out of $- h^{2}$ .

$\frac{h (- h) - 2 h t + 2 h}{h}$

Step 5.1.9.2

Factor $h$ out of $- 2 h t$ .

$\frac{h (- h) + h (- 2 t) + 2 h}{h}$

Step 5.1.9.3

Factor $h$ out of $2 h$ .

$\frac{h (- h) + h (- 2 t) + h \cdot 2}{h}$

Step 5.1.9.4

Factor $h$ out of $h (- h) + h (- 2 t)$ .

$\frac{h (- h - 2 t) + h \cdot 2}{h}$

Step 5.1.9.5

Factor $h$ out of $h (- h - 2 t) + h \cdot 2$ .

$\frac{h (- h - 2 t + 2)}{h}$

Step 5.2

Reduce the expression by cancelling the common factors.

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Step 5.2.1

Cancel the common factor of $h$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{h (- h - 2 t + 2)}{h}$

Step 5.2.1.2

Divide $- h - 2 t + 2$ by $1$ .

$- h - 2 t + 2$

Step 5.2.2

Reorder $- h$ and $- 2 t$ .

$- 2 t - h + 2$

Step 6