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