Precalculus Examples

$f (x) = - 2 t^{2} + 4 t + 1$

Step 1

Write $f (x) = - 2 t^{2} + 4 t + 1$ as an equation.

$y = - 2 t^{2} + 4 t + 1$

Step 2

Rewrite the equation in vertex form.

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Step 2.1

Complete the square for $- 2 t^{2} + 4 t + 1$ .

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Step 2.1.1

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

$a = - 2$

$b = 4$

$c = 1$

Step 2.1.2

Consider the vertex form of a parabola.

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

Step 2.1.3

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

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Step 2.1.3.1

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

$d = \frac{4}{2 \cdot - 2}$

Step 2.1.3.2

Simplify the right side.

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Step 2.1.3.2.1

Cancel the common factor of $4$ and $2$ .

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Step 2.1.3.2.1.1

Factor $2$ out of $4$ .

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

Step 2.1.3.2.1.2

Cancel the common factors.

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Step 2.1.3.2.1.2.1

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

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

Step 2.1.3.2.1.2.2

Cancel the common factor.

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

Step 2.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{2}{- 2}$

Step 2.1.3.2.2

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

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Step 2.1.3.2.2.1

Factor $2$ out of $2$ .

$d = \frac{2 (1)}{- 2}$

Step 2.1.3.2.2.2

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

$d = - 1 \cdot 1$

Step 2.1.3.2.3

Multiply $- 1$ by $1$ .

$d = - 1$

Step 2.1.4

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

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Step 2.1.4.1

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

$e = 1 - \frac{4^{2}}{4 \cdot - 2}$

Step 2.1.4.2

Simplify the right side.

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Step 2.1.4.2.1

Simplify each term.

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Step 2.1.4.2.1.1

Cancel the common factor of $4^{2}$ and $4$ .

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Step 2.1.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = 1 - \frac{4 \cdot 4}{4 \cdot - 2}$

Step 2.1.4.2.1.1.2

Cancel the common factors.

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Step 2.1.4.2.1.1.2.1

Factor $4$ out of $4 \cdot - 2$ .

$e = 1 - \frac{4 \cdot 4}{4 (- 2)}$

Step 2.1.4.2.1.1.2.2

Cancel the common factor.

$e = 1 - \frac{4 \cdot 4}{4 \cdot - 2}$

Step 2.1.4.2.1.1.2.3

Rewrite the expression.

$e = 1 - \frac{4}{- 2}$

Step 2.1.4.2.1.2

Cancel the common factor of $4$ and $- 2$ .

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Step 2.1.4.2.1.2.1

Factor $2$ out of $4$ .

$e = 1 - \frac{2 (2)}{- 2}$

Step 2.1.4.2.1.2.2

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$e = 1 - (- 1 \cdot 2)$

Step 2.1.4.2.1.3

Multiply $- (- 1 \cdot 2)$ .

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Step 2.1.4.2.1.3.1

Multiply $- 1$ by $2$ .

$e = 1 - - 2$

Step 2.1.4.2.1.3.2

Multiply $- 1$ by $- 2$ .

$e = 1 + 2$

Step 2.1.4.2.2

Add $1$ and $2$ .

$e = 3$

Step 2.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(t - 1)}^{2} + 3$ .

$- 2 {(t - 1)}^{2} + 3$

Step 2.2

Set $y$ equal to the new right side.

$y = - 2 {(t - 1)}^{2} + 3$

Step 3

Use the vertex form, $y = a {(t - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 2$

$h = 1$

$k = 3$

Step 4

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(1, 3)$

Step 6

Find $p$ , the distance from the vertex to the focus.

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Step 6.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 6.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 2}$

Step 6.3

Simplify.

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Step 6.3.1

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Step 6.3.2

Move the negative in front of the fraction.

$- \frac{1}{8}$

Step 7

Find the focus.

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Step 7.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 7.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(1, \frac{23}{8})$

Step 8

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$t = 1$

Step 9

Find the directrix.

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Step 9.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 9.2

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = \frac{25}{8}$

Step 10

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: $(1, 3)$

Focus: $(1, \frac{23}{8})$

Axis of Symmetry: $t = 1$

Directrix: $y = \frac{25}{8}$

Step 11