Graph y=1/4(x-5)^2

$y = \frac{1}{4} {(x - 5)}^{2}$

Step 1

Simplify $\frac{1}{4} \cdot {(x - 5)}^{2}$ .

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$y = \frac{1}{4} \cdot ((x - 5) (x - 5))$

Expand $(x - 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{1}{4} \cdot (x (x - 5) - 5 (x - 5))$

Apply the distributive property.

$y = \frac{1}{4} \cdot (x \cdot x + x \cdot - 5 - 5 (x - 5))$

Apply the distributive property.

$y = \frac{1}{4} \cdot (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = \frac{1}{4} \cdot (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5)$

Move $- 5$ to the left of $x$ .

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

Multiply $- 5$ by $- 5$ .

$y = \frac{1}{4} \cdot (x^{2} - 5 x - 5 x + 25)$

Subtract $5 x$ from $- 5 x$ .

$y = \frac{1}{4} \cdot (x^{2} - 10 x + 25)$

Apply the distributive property.

$y = \frac{1}{4} x^{2} + \frac{1}{4} (- 10 x) + \frac{1}{4} \cdot 25$

Simplify.

Combine $\frac{1}{4}$ and $x^{2}$ .

$y = \frac{x^{2}}{4} + \frac{1}{4} (- 10 x) + \frac{1}{4} \cdot 25$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$y = \frac{x^{2}}{4} + \frac{1}{2 (2)} (- 10 x) + \frac{1}{4} \cdot 25$

Factor $2$ out of $- 10 x$ .

$y = \frac{x^{2}}{4} + \frac{1}{2 (2)} (2 (- 5 x)) + \frac{1}{4} \cdot 25$

Cancel the common factor.

$y = \frac{x^{2}}{4} + \frac{1}{2 \cdot 2} (2 (- 5 x)) + \frac{1}{4} \cdot 25$

Rewrite the expression.

$y = \frac{x^{2}}{4} + \frac{1}{2} (- 5 x) + \frac{1}{4} \cdot 25$

Combine $- 5$ and $\frac{1}{2}$ .

$y = \frac{x^{2}}{4} + \frac{- 5}{2} x + \frac{1}{4} \cdot 25$

Combine $\frac{- 5}{2}$ and $x$ .

$y = \frac{x^{2}}{4} + \frac{- 5 x}{2} + \frac{1}{4} \cdot 25$

Combine $\frac{1}{4}$ and $25$ .

$y = \frac{x^{2}}{4} + \frac{- 5 x}{2} + \frac{25}{4}$

Move the negative in front of the fraction.

$y = \frac{x^{2}}{4} - \frac{5 x}{2} + \frac{25}{4}$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $\frac{x^{2}}{4} - \frac{5 x}{2} + \frac{25}{4}$ .

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

$a = \frac{1}{4}$

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

$c = \frac{25}{4}$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- \frac{5}{2}}{2 (\frac{1}{4})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Combine $2$ and $\frac{1}{4}$ .

$d = - \frac{5}{2} \cdot \frac{1}{\frac{2}{4}}$

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{5}{2}$ into the numerator.

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

Factor $2$ out of $1 \cdot 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = - 5$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{25}{4} - \frac{{(- \frac{5}{2})}^{2}}{4 (\frac{1}{4})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{5}{2}$ .

$e = \frac{25}{4} - \frac{{(- 1)}^{2} {(\frac{5}{2})}^{2}}{4 (\frac{1}{4})}$

Raise $- 1$ to the power of $2$ .

$e = \frac{25}{4} - \frac{1 {(\frac{5}{2})}^{2}}{4 (\frac{1}{4})}$

Apply the product rule to $\frac{5}{2}$ .

$e = \frac{25}{4} - \frac{1 \frac{5^{2}}{2^{2}}}{4 (\frac{1}{4})}$

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

$e = \frac{25}{4} - \frac{1 \frac{25}{2^{2}}}{4 (\frac{1}{4})}$

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

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

Multiply $\frac{25}{4}$ by $1$ .

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

Combine $4$ and $\frac{1}{4}$ .

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

Divide $4$ by $4$ .

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

Divide $\frac{25}{4}$ by $1$ .

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

Combine the numerators over the common denominator.

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

Subtract $25$ from $25$ .

$e = \frac{0}{4}$

Divide $0$ by $4$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(x - 5)}^{2} + 0$ .

$\frac{1}{4} {(x - 5)}^{2} + 0$

Set $y$ equal to the new right side.

$y = \frac{1}{4} \cdot {(x - 5)}^{2} + 0$

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

$a = \frac{1}{4}$

$h = 5$

$k = 0$

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

Opens Up

Find the vertex $(h, k)$ .

$(5, 0)$

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.

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

Simplify.

Combine $4$ and $\frac{1}{4}$ .

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

Simplify by dividing numbers.

Divide $4$ by $4$ .

$\frac{1}{1}$

Divide $1$ by $1$ .

$1$

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.

$(5, 1)$

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

$x = 5$

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.

$y = - 1$

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

Direction: Opens Up

Vertex: $(5, 0)$

Focus: $(5, 1)$

Axis of Symmetry: $x = 5$

Directrix: $y = - 1$

Direction: Opens Up

Vertex: $(5, 0)$

Focus: $(5, 1)$

Axis of Symmetry: $x = 5$

Directrix: $y = - 1$

Step 3

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{{(3)}^{2}}{4} - \frac{5 (3)}{2} + \frac{25}{4}$

Simplify the result.

Combine fractions.

Combine the numerators over the common denominator.

$f (3) = \frac{{(3)}^{2} + 25}{4} + \frac{- 5 \cdot 3}{2}$

Simplify the expression.

Raise $3$ to the power of $2$ .

$f (3) = \frac{9 + 25}{4} + \frac{- 5 \cdot 3}{2}$

Add $9$ and $25$ .

$f (3) = \frac{34}{4} + \frac{- 5 \cdot 3}{2}$

Multiply $- 5$ by $3$ .

$f (3) = \frac{34}{4} + \frac{- 15}{2}$

Simplify each term.

Cancel the common factor of $34$ and $4$ .

Factor $2$ out of $34$ .

$f (3) = \frac{2 (17)}{4} + \frac{- 15}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (3) = \frac{2 \cdot 17}{2 \cdot 2} + \frac{- 15}{2}$

Cancel the common factor.

$f (3) = \frac{2 \cdot 17}{2 \cdot 2} + \frac{- 15}{2}$

Rewrite the expression.

$f (3) = \frac{17}{2} + \frac{- 15}{2}$

Move the negative in front of the fraction.

$f (3) = \frac{17}{2} - \frac{15}{2}$

Combine fractions.

Combine the numerators over the common denominator.

$f (3) = \frac{17 - 15}{2}$

Simplify the expression.

Subtract $15$ from $17$ .

$f (3) = \frac{2}{2}$

Divide $2$ by $2$ .

$f (3) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = 3$ is $1$ .

$y = 1$

Replace the variable $x$ with $4$ in the expression.

$f (4) = \frac{{(4)}^{2}}{4} - \frac{5 (4)}{2} + \frac{25}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (4) = \frac{{(4)}^{2} + 25}{4} + \frac{- 5 \cdot 4}{2}$

Simplify the expression.

Raise $4$ to the power of $2$ .

$f (4) = \frac{16 + 25}{4} + \frac{- 5 \cdot 4}{2}$

Add $16$ and $25$ .

$f (4) = \frac{41}{4} + \frac{- 5 \cdot 4}{2}$

Multiply $- 5$ by $4$ .

$f (4) = \frac{41}{4} + \frac{- 20}{2}$

Divide $- 20$ by $2$ .

$f (4) = \frac{41}{4} - 10$

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (4) = \frac{41}{4} - 10 \cdot \frac{4}{4}$

Combine $- 10$ and $\frac{4}{4}$ .

$f (4) = \frac{41}{4} + \frac{- 10 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (4) = \frac{41 - 10 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 10$ by $4$ .

$f (4) = \frac{41 - 40}{4}$

Subtract $40$ from $41$ .

$f (4) = \frac{1}{4}$

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

The $y$ value at $x = 4$ is $\frac{1}{4}$ .

$y = \frac{1}{4}$

Replace the variable $x$ with $7$ in the expression.

$f (7) = \frac{{(7)}^{2}}{4} - \frac{5 (7)}{2} + \frac{25}{4}$

Simplify the result.

Combine fractions.

Combine the numerators over the common denominator.

$f (7) = \frac{{(7)}^{2} + 25}{4} + \frac{- 5 \cdot 7}{2}$

Simplify the expression.

Raise $7$ to the power of $2$ .

$f (7) = \frac{49 + 25}{4} + \frac{- 5 \cdot 7}{2}$

Add $49$ and $25$ .

$f (7) = \frac{74}{4} + \frac{- 5 \cdot 7}{2}$

Multiply $- 5$ by $7$ .

$f (7) = \frac{74}{4} + \frac{- 35}{2}$

Simplify each term.

Cancel the common factor of $74$ and $4$ .

Factor $2$ out of $74$ .

$f (7) = \frac{2 (37)}{4} + \frac{- 35}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (7) = \frac{2 \cdot 37}{2 \cdot 2} + \frac{- 35}{2}$

Cancel the common factor.

$f (7) = \frac{2 \cdot 37}{2 \cdot 2} + \frac{- 35}{2}$

Rewrite the expression.

$f (7) = \frac{37}{2} + \frac{- 35}{2}$

Move the negative in front of the fraction.

$f (7) = \frac{37}{2} - \frac{35}{2}$

Combine fractions.

Combine the numerators over the common denominator.

$f (7) = \frac{37 - 35}{2}$

Simplify the expression.

Subtract $35$ from $37$ .

$f (7) = \frac{2}{2}$

Divide $2$ by $2$ .

$f (7) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = 7$ is $1$ .

$y = 1$

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{{(6)}^{2}}{4} - \frac{5 (6)}{2} + \frac{25}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (6) = \frac{{(6)}^{2} + 25}{4} + \frac{- 5 \cdot 6}{2}$

Simplify the expression.

Raise $6$ to the power of $2$ .

$f (6) = \frac{36 + 25}{4} + \frac{- 5 \cdot 6}{2}$

Add $36$ and $25$ .

$f (6) = \frac{61}{4} + \frac{- 5 \cdot 6}{2}$

Multiply $- 5$ by $6$ .

$f (6) = \frac{61}{4} + \frac{- 30}{2}$

Divide $- 30$ by $2$ .

$f (6) = \frac{61}{4} - 15$

To write $- 15$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (6) = \frac{61}{4} - 15 \cdot \frac{4}{4}$

Combine $- 15$ and $\frac{4}{4}$ .

$f (6) = \frac{61}{4} + \frac{- 15 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (6) = \frac{61 - 15 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 15$ by $4$ .

$f (6) = \frac{61 - 60}{4}$

Subtract $60$ from $61$ .

$f (6) = \frac{1}{4}$

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

The $y$ value at $x = 6$ is $\frac{1}{4}$ .

$y = \frac{1}{4}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 3 & 1 \\ 4 & \frac{1}{4} \\ 5 & 0 \\ 6 & \frac{1}{4} \\ 7 & 1 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(5, 0)$

Focus: $(5, 1)$

Axis of Symmetry: $x = 5$

Directrix: $y = - 1$

$\begin{array}{c} x & y \\ 3 & 1 \\ 4 & \frac{1}{4} \\ 5 & 0 \\ 6 & \frac{1}{4} \\ 7 & 1 \end{array}$

Step 5

Graph y=1/4(x-5)^2

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