Find the Properties y=(x-2)^2+5

$y = {(x - 2)}^{2} + 5$

Step 1

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

$a = 1$

$h = 2$

$k = 5$

Step 2

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 3

Find the vertex $(h, k)$ .

$(2, 5)$

Step 4

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

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

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

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

Cancel the common factor of $1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{4}$

Step 5

Find the focus.

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

$(h, k + p)$

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

$(2, \frac{21}{4})$

Step 6

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

$x = 2$

Step 7

Find the directrix.

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.

$y = k - p$

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

$y = \frac{19}{4}$

Step 8

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

Direction: Opens Up

Vertex: $(2, 5)$

Focus: $(2, \frac{21}{4})$

Axis of Symmetry: $x = 2$

Directrix: $y = \frac{19}{4}$

Step 9

Find the Properties y=(x-2)^2+5

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