Finite Math Examples

$s (t) = 95 - 16 t^{2}$

Step 1

The parent function is the simplest form of the type of function given.

$g (t) = t^{2}$

Step 2

The transformation being described is from $g (t) = t^{2}$ to $s (t) = 95 - 16 t^{2}$ .

$g (t) = t^{2} \to s (t) = 95 - 16 t^{2}$

Step 3

Find the vertex form of $s (t) = 95 - 16 t^{2}$ .

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Step 3.1

Reorder $95$ and $- 16 x^{2}$ .

$y = - 16 x^{2} + 95$

Step 3.2

Complete the square for $- 16 x^{2} + 95$ .

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Step 3.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 16$

$b = 0$

$c = 95$

Step 3.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 3.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 3.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 \cdot - 16}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 3.2.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot - 16}$

Step 3.2.3.2.1.2

Cancel the common factors.

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Step 3.2.3.2.1.2.1

Factor $2$ out of $2 \cdot - 16$ .

$d = \frac{2 (0)}{2 (- 16)}$

Step 3.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot - 16}$

Step 3.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{0}{- 16}$

Step 3.2.3.2.2

Cancel the common factor of $0$ and $- 16$ .

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Step 3.2.3.2.2.1

Factor $16$ out of $0$ .

$d = \frac{16 (0)}{- 16}$

Step 3.2.3.2.2.2

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Step 3.2.3.2.3

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Step 3.2.3.2.4

Multiply $- 1$ by $0$ .

$d = 0$

Step 3.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 95 - \frac{0^{2}}{4 \cdot - 16}$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify each term.

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Step 3.2.4.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = 95 - \frac{0}{4 \cdot - 16}$

Step 3.2.4.2.1.2

Multiply $4$ by $- 16$ .

$e = 95 - \frac{0}{- 64}$

Step 3.2.4.2.1.3

Divide $0$ by $- 64$ .

$e = 95 - 0$

Step 3.2.4.2.1.4

Multiply $- 1$ by $0$ .

$e = 95 + 0$

Step 3.2.4.2.2

Add $95$ and $0$ .

$e = 95$

Step 3.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 16 {(x + 0)}^{2} + 95$ .

$- 16 {(x + 0)}^{2} + 95$

Step 3.3

Set $y$ equal to the new right side.

$y = - 16 {(x + 0)}^{2} + 95$

Step 4

The horizontal shift depends on the value of $h$ . The horizontal shift is described as:

$s (t) = f (x + h)$ - The graph is shifted to the left $h$ units.

$s (t) = f (x - h)$ - The graph is shifted to the right $h$ units.

In this case, $h = 0$ which means that the graph is not shifted to the left or right.

Horizontal Shift: None

Step 5

The vertical shift depends on the value of $k$ . The vertical shift is described as:

$s (t) = f (x) + k$ - The graph is shifted up $k$ units.

$s (t) = f (x) - k$ - The graph is shifted down $k$ units.

Vertical Shift: Up $95$ Units

Step 6

The graph is reflected about the x-axis when $s (t) = - f (x)$ .

Reflection about the x-axis: Reflected

Step 7

The graph is reflected about the y-axis when $s (t) = f (- x)$ .

Reflection about the y-axis: None

Step 8

Compressing and stretching depends on the value of $a$ .

When $a$ is greater than $1$ : Vertically stretched

When $a$ is between $0$ and $1$ : Vertically compressed

Vertical Compression or Stretch: Stretched

Step 9

Compare and list the transformations.

Parent Function: $g (t) = t^{2}$

Horizontal Shift: None

Vertical Shift: Up $95$ Units

Reflection about the x-axis: Reflected

Reflection about the y-axis: None

Vertical Compression or Stretch: Stretched

Step 10