Find the Properties f(x)=x^2-5x+1

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

Step 1

Write $f (x) = x^{2} - 5 x + 1$ as an equation.

$y = x^{2} - 5 x + 1$

Step 2

Rewrite the equation in vertex form.

Complete the square for $x^{2} - 5 x + 1$ .

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

$a = 1$

$b = - 5$

$c = 1$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{- 5}{2 \cdot 1}$

Simplify the right side.

Multiply $2$ by $1$ .

$d = \frac{- 5}{2}$

Move the negative in front of the fraction.

$d = - \frac{5}{2}$

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

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

$e = 1 - \frac{{(- 5)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 5$ to the power of $2$ .

$e = 1 - \frac{25}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 1 - \frac{25}{4}$

Write $1$ as a fraction with a common denominator.

$e = \frac{4}{4} - \frac{25}{4}$

Combine the numerators over the common denominator.

$e = \frac{4 - 25}{4}$

Subtract $25$ from $4$ .

$e = \frac{- 21}{4}$

Move the negative in front of the fraction.

$e = - \frac{21}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{5}{2})}^{2} - \frac{21}{4}$ .

${(x - \frac{5}{2})}^{2} - \frac{21}{4}$

Set $y$ equal to the new right side.

$y = {(x - \frac{5}{2})}^{2} - \frac{21}{4}$

Step 3

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

$a = 1$

$h = \frac{5}{2}$

$k = - \frac{21}{4}$

Step 4

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

Opens Up

Step 5

Find the vertex $(h, k)$ .

$(\frac{5}{2}, - \frac{21}{4})$

Step 6

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 7

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.

$(\frac{5}{2}, - 5)$

Step 8

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

$x = \frac{5}{2}$

Step 9

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.

$y = - \frac{11}{2}$

Step 10

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

Direction: Opens Up

Vertex: $(\frac{5}{2}, - \frac{21}{4})$

Focus: $(\frac{5}{2}, - 5)$

Axis of Symmetry: $x = \frac{5}{2}$

Directrix: $y = - \frac{11}{2}$

Step 11

Find the Properties f(x)=x^2-5x+1

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