Graph 2x^2-12x+28y=38

$2 x^{2} - 12 x + 28 y = 38$

Step 1

Solve for $y$ .

Move all terms not containing $y$ to the right side of the equation.

Subtract $2 x^{2}$ from both sides of the equation.

$- 12 x + 28 y = 38 - 2 x^{2}$

Add $12 x$ to both sides of the equation.

$28 y = 38 - 2 x^{2} + 12 x$

Divide each term in $28 y = 38 - 2 x^{2} + 12 x$ by $28$ and simplify.

Divide each term in $28 y = 38 - 2 x^{2} + 12 x$ by $28$ .

$\frac{28 y}{28} = \frac{38}{28} + \frac{- 2 x^{2}}{28} + \frac{12 x}{28}$

Simplify the left side.

Cancel the common factor of $28$ .

Cancel the common factor.

$\frac{28 y}{28} = \frac{38}{28} + \frac{- 2 x^{2}}{28} + \frac{12 x}{28}$

Divide $y$ by $1$ .

$y = \frac{38}{28} + \frac{- 2 x^{2}}{28} + \frac{12 x}{28}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $38$ and $28$ .

Factor $2$ out of $38$ .

$y = \frac{2 (19)}{28} + \frac{- 2 x^{2}}{28} + \frac{12 x}{28}$

Cancel the common factors.

Factor $2$ out of $28$ .

$y = \frac{2 \cdot 19}{2 \cdot 14} + \frac{- 2 x^{2}}{28} + \frac{12 x}{28}$

Cancel the common factor.

$y = \frac{2 \cdot 19}{2 \cdot 14} + \frac{- 2 x^{2}}{28} + \frac{12 x}{28}$

Rewrite the expression.

$y = \frac{19}{14} + \frac{- 2 x^{2}}{28} + \frac{12 x}{28}$

Cancel the common factor of $- 2$ and $28$ .

Factor $2$ out of $- 2 x^{2}$ .

$y = \frac{19}{14} + \frac{2 (- x^{2})}{28} + \frac{12 x}{28}$

Cancel the common factors.

Factor $2$ out of $28$ .

$y = \frac{19}{14} + \frac{2 (- x^{2})}{2 (14)} + \frac{12 x}{28}$

Cancel the common factor.

$y = \frac{19}{14} + \frac{2 (- x^{2})}{2 \cdot 14} + \frac{12 x}{28}$

Rewrite the expression.

$y = \frac{19}{14} + \frac{- x^{2}}{14} + \frac{12 x}{28}$

Move the negative in front of the fraction.

$y = \frac{19}{14} - \frac{x^{2}}{14} + \frac{12 x}{28}$

Cancel the common factor of $12$ and $28$ .

Factor $4$ out of $12 x$ .

$y = \frac{19}{14} - \frac{x^{2}}{14} + \frac{4 (3 x)}{28}$

Cancel the common factors.

Factor $4$ out of $28$ .

$y = \frac{19}{14} - \frac{x^{2}}{14} + \frac{4 (3 x)}{4 (7)}$

Cancel the common factor.

$y = \frac{19}{14} - \frac{x^{2}}{14} + \frac{4 (3 x)}{4 \cdot 7}$

Rewrite the expression.

$y = \frac{19}{14} - \frac{x^{2}}{14} + \frac{3 x}{7}$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Move $\frac{19}{14}$ .

$y = - \frac{x^{2}}{14} + \frac{3 x}{7} + \frac{19}{14}$

Complete the square for $- \frac{x^{2}}{14} + \frac{3 x}{7} + \frac{19}{14}$ .

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

$a = - \frac{1}{14}$

$b = \frac{3}{7}$

$c = \frac{19}{14}$

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

$d = \frac{\frac{3}{7}}{2 (- \frac{1}{14})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{7} \cdot \frac{1}{2 (- \frac{1}{14})}$

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$d = \frac{3}{7} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{14})}$

Move the negative in front of the fraction.

$d = \frac{3}{7} (- \frac{1}{2 (\frac{1}{14})})$

Combine $2$ and $\frac{1}{14}$ .

$d = \frac{3}{7} (- \frac{1}{\frac{2}{14}})$

Cancel the common factor of $2$ and $14$ .

Factor $2$ out of $2$ .

$d = \frac{3}{7} (- \frac{1}{\frac{2 (1)}{14}})$

Cancel the common factors.

Factor $2$ out of $14$ .

$d = \frac{3}{7} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 7}})$

Cancel the common factor.

$d = \frac{3}{7} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 7}})$

Rewrite the expression.

$d = \frac{3}{7} (- \frac{1}{\frac{1}{7}})$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{7} (- (1 \cdot 7))$

Cancel the common factor of $7$ .

Factor $7$ out of $- (1 \cdot 7)$ .

$d = \frac{3}{7} (7 (- 1 \cdot (1)))$

Cancel the common factor.

$d = \frac{3}{7} (7 (- 1 \cdot (1)))$

Rewrite the expression.

$d = 3 (- 1 \cdot (1))$

Multiply $- 1$ by $1$ .

$d = 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$d = - 3$

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

$e = \frac{19}{14} - \frac{{(\frac{3}{7})}^{2}}{4 (- \frac{1}{14})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{3}{7}$ .

$e = \frac{19}{14} - \frac{\frac{3^{2}}{7^{2}}}{4 (- \frac{1}{14})}$

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

$e = \frac{19}{14} - \frac{\frac{9}{7^{2}}}{4 (- \frac{1}{14})}$

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

$e = \frac{19}{14} - \frac{\frac{9}{49}}{4 (- \frac{1}{14})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = \frac{19}{14} - \frac{\frac{9}{49}}{- 4 (\frac{1}{14})}$

Combine $- 4$ and $\frac{1}{14}$ .

$e = \frac{19}{14} - \frac{\frac{9}{49}}{\frac{- 4}{14}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $14$ .

Factor $2$ out of $- 4$ .

$e = \frac{19}{14} - \frac{\frac{9}{49}}{\frac{2 (- 2)}{14}}$

Cancel the common factors.

Factor $2$ out of $14$ .

$e = \frac{19}{14} - \frac{\frac{9}{49}}{\frac{2 \cdot - 2}{2 \cdot 7}}$

Cancel the common factor.

$e = \frac{19}{14} - \frac{\frac{9}{49}}{\frac{2 \cdot - 2}{2 \cdot 7}}$

Rewrite the expression.

$e = \frac{19}{14} - \frac{\frac{9}{49}}{\frac{- 2}{7}}$

Move the negative in front of the fraction.

$e = \frac{19}{14} - \frac{\frac{9}{49}}{- \frac{2}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{19}{14} - (\frac{9}{49} (- \frac{7}{2}))$

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$e = \frac{19}{14} - (\frac{9}{49} \cdot \frac{- 7}{2})$

Factor $7$ out of $49$ .

$e = \frac{19}{14} - (\frac{9}{7 (7)} \cdot \frac{- 7}{2})$

Factor $7$ out of $- 7$ .

$e = \frac{19}{14} - (\frac{9}{7 \cdot 7} \cdot \frac{7 \cdot - 1}{2})$

Cancel the common factor.

$e = \frac{19}{14} - (\frac{9}{7 \cdot 7} \cdot \frac{7 \cdot - 1}{2})$

Rewrite the expression.

$e = \frac{19}{14} - (\frac{9}{7} \cdot \frac{- 1}{2})$

Multiply $\frac{9}{7}$ by $\frac{- 1}{2}$ .

$e = \frac{19}{14} - \frac{9 \cdot - 1}{7 \cdot 2}$

Multiply $9$ by $- 1$ .

$e = \frac{19}{14} - \frac{- 9}{7 \cdot 2}$

Multiply $7$ by $2$ .

$e = \frac{19}{14} - \frac{- 9}{14}$

Move the negative in front of the fraction.

$e = \frac{19}{14} - - \frac{9}{14}$

Multiply $- - \frac{9}{14}$ .

Multiply $- 1$ by $- 1$ .

$e = \frac{19}{14} + 1 (\frac{9}{14})$

Multiply $\frac{9}{14}$ by $1$ .

$e = \frac{19}{14} + \frac{9}{14}$

Combine the numerators over the common denominator.

$e = \frac{19 + 9}{14}$

Add $19$ and $9$ .

$e = \frac{28}{14}$

Divide $28$ by $14$ .

$e = 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{14} {(x - 3)}^{2} + 2$ .

$- \frac{1}{14} {(x - 3)}^{2} + 2$

Set $y$ equal to the new right side.

$y = - \frac{1}{14} \cdot {(x - 3)}^{2} + 2$

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

$a = - \frac{1}{14}$

$h = 3$

$k = 2$

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

Opens Down

Find the vertex $(h, k)$ .

$(3, 2)$

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.

$\frac{1}{4 \cdot (- \frac{1}{14})}$

Simplify.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{14})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{14})}$

Combine $4$ and $\frac{1}{14}$ .

$- \frac{1}{\frac{4}{14}}$

Cancel the common factor of $4$ and $14$ .

Factor $2$ out of $4$ .

$- \frac{1}{\frac{2 (2)}{14}}$

Cancel the common factors.

Factor $2$ out of $14$ .

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 7}}$

Cancel the common factor.

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 7}}$

Rewrite the expression.

$- \frac{1}{\frac{2}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{7}{2}))$

Multiply $\frac{7}{2}$ by $1$ .

$- \frac{7}{2}$

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.

$(3, - \frac{3}{2})$

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

$x = 3$

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.

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

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

Direction: Opens Down

Vertex: $(3, 2)$

Focus: $(3, - \frac{3}{2})$

Axis of Symmetry: $x = 3$

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

Direction: Opens Down

Vertex: $(3, 2)$

Focus: $(3, - \frac{3}{2})$

Axis of Symmetry: $x = 3$

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

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 $2$ in the expression.

$f (2) = - \frac{{(2)}^{2}}{14} + \frac{3 (2)}{7} + \frac{19}{14}$

Simplify the result.

Combine the numerators over the common denominator.

$f (2) = \frac{- {(2)}^{2} + 19}{14} + \frac{3 (2)}{7}$

Simplify each term.

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

$f (2) = \frac{- 1 \cdot 4 + 19}{14} + \frac{3 (2)}{7}$

Multiply $- 1$ by $4$ .

$f (2) = \frac{- 4 + 19}{14} + \frac{3 (2)}{7}$

Simplify the expression.

Add $- 4$ and $19$ .

$f (2) = \frac{15}{14} + \frac{3 (2)}{7}$

Multiply $3$ by $2$ .

$f (2) = \frac{15}{14} + \frac{6}{7}$

To write $\frac{6}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (2) = \frac{15}{14} + \frac{6}{7} \cdot \frac{2}{2}$

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{6}{7}$ by $\frac{2}{2}$ .

$f (2) = \frac{15}{14} + \frac{6 \cdot 2}{7 \cdot 2}$

Multiply $7$ by $2$ .

$f (2) = \frac{15}{14} + \frac{6 \cdot 2}{14}$

Combine the numerators over the common denominator.

$f (2) = \frac{15 + 6 \cdot 2}{14}$

Simplify the numerator.

Multiply $6$ by $2$ .

$f (2) = \frac{15 + 12}{14}$

Add $15$ and $12$ .

$f (2) = \frac{27}{14}$

The final answer is $\frac{27}{14}$ .

$\frac{27}{14}$

The $y$ value at $x = 2$ is $\frac{27}{14}$ .

$y = \frac{27}{14}$

Replace the variable $x$ with $1$ in the expression.

$f (1) = - \frac{{(1)}^{2}}{14} + \frac{3 (1)}{7} + \frac{19}{14}$

Simplify the result.

Combine the numerators over the common denominator.

$f (1) = \frac{- {(1)}^{2} + 19}{14} + \frac{3 (1)}{7}$

Simplify each term.

One to any power is one.

$f (1) = \frac{- 1 \cdot 1 + 19}{14} + \frac{3 (1)}{7}$

Multiply $- 1$ by $1$ .

$f (1) = \frac{- 1 + 19}{14} + \frac{3 (1)}{7}$

Simplify terms.

Add $- 1$ and $19$ .

$f (1) = \frac{18}{14} + \frac{3 (1)}{7}$

Multiply $3$ by $1$ .

$f (1) = \frac{18}{14} + \frac{3}{7}$

Cancel the common factor of $18$ and $14$ .

Factor $2$ out of $18$ .

$f (1) = \frac{2 (9)}{14} + \frac{3}{7}$

Cancel the common factors.

Factor $2$ out of $14$ .

$f (1) = \frac{2 \cdot 9}{2 \cdot 7} + \frac{3}{7}$

Cancel the common factor.

$f (1) = \frac{2 \cdot 9}{2 \cdot 7} + \frac{3}{7}$

Rewrite the expression.

$f (1) = \frac{9}{7} + \frac{3}{7}$

Combine the numerators over the common denominator.

$f (1) = \frac{9 + 3}{7}$

Add $9$ and $3$ .

$f (1) = \frac{12}{7}$

The final answer is $\frac{12}{7}$ .

$\frac{12}{7}$

The $y$ value at $x = 1$ is $\frac{12}{7}$ .

$y = \frac{12}{7}$

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

$f (4) = - \frac{{(4)}^{2}}{14} + \frac{3 (4)}{7} + \frac{19}{14}$

Simplify the result.

Combine the numerators over the common denominator.

$f (4) = \frac{- {(4)}^{2} + 19}{14} + \frac{3 (4)}{7}$

Simplify each term.

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

$f (4) = \frac{- 1 \cdot 16 + 19}{14} + \frac{3 (4)}{7}$

Multiply $- 1$ by $16$ .

$f (4) = \frac{- 16 + 19}{14} + \frac{3 (4)}{7}$

Simplify the expression.

Add $- 16$ and $19$ .

$f (4) = \frac{3}{14} + \frac{3 (4)}{7}$

Multiply $3$ by $4$ .

$f (4) = \frac{3}{14} + \frac{12}{7}$

To write $\frac{12}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (4) = \frac{3}{14} + \frac{12}{7} \cdot \frac{2}{2}$

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{12}{7}$ by $\frac{2}{2}$ .

$f (4) = \frac{3}{14} + \frac{12 \cdot 2}{7 \cdot 2}$

Multiply $7$ by $2$ .

$f (4) = \frac{3}{14} + \frac{12 \cdot 2}{14}$

Combine the numerators over the common denominator.

$f (4) = \frac{3 + 12 \cdot 2}{14}$

Simplify the numerator.

Multiply $12$ by $2$ .

$f (4) = \frac{3 + 24}{14}$

Add $3$ and $24$ .

$f (4) = \frac{27}{14}$

The final answer is $\frac{27}{14}$ .

$\frac{27}{14}$

The $y$ value at $x = 4$ is $\frac{27}{14}$ .

$y = \frac{27}{14}$

Replace the variable $x$ with $5$ in the expression.

$f (5) = - \frac{{(5)}^{2}}{14} + \frac{3 (5)}{7} + \frac{19}{14}$

Simplify the result.

Combine the numerators over the common denominator.

$f (5) = \frac{- {(5)}^{2} + 19}{14} + \frac{3 (5)}{7}$

Simplify each term.

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

$f (5) = \frac{- 1 \cdot 25 + 19}{14} + \frac{3 (5)}{7}$

Multiply $- 1$ by $25$ .

$f (5) = \frac{- 25 + 19}{14} + \frac{3 (5)}{7}$

Simplify the expression.

Add $- 25$ and $19$ .

$f (5) = \frac{- 6}{14} + \frac{3 (5)}{7}$

Multiply $3$ by $5$ .

$f (5) = \frac{- 6}{14} + \frac{15}{7}$

Simplify each term.

Cancel the common factor of $- 6$ and $14$ .

Factor $2$ out of $- 6$ .

$f (5) = \frac{2 (- 3)}{14} + \frac{15}{7}$

Cancel the common factors.

Factor $2$ out of $14$ .

$f (5) = \frac{2 \cdot - 3}{2 \cdot 7} + \frac{15}{7}$

Cancel the common factor.

$f (5) = \frac{2 \cdot - 3}{2 \cdot 7} + \frac{15}{7}$

Rewrite the expression.

$f (5) = \frac{- 3}{7} + \frac{15}{7}$

Move the negative in front of the fraction.

$f (5) = - \frac{3}{7} + \frac{15}{7}$

Combine fractions.

Combine the numerators over the common denominator.

$f (5) = \frac{- 3 + 15}{7}$

Add $- 3$ and $15$ .

$f (5) = \frac{12}{7}$

The final answer is $\frac{12}{7}$ .

$\frac{12}{7}$

The $y$ value at $x = 5$ is $\frac{12}{7}$ .

$y = \frac{12}{7}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & \frac{12}{7} \\ 2 & \frac{27}{14} \\ 3 & 2 \\ 4 & \frac{27}{14} \\ 5 & \frac{12}{7} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(3, 2)$

Focus: $(3, - \frac{3}{2})$

Axis of Symmetry: $x = 3$

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

$\begin{array}{c} x & y \\ 1 & \frac{12}{7} \\ 2 & \frac{27}{14} \\ 3 & 2 \\ 4 & \frac{27}{14} \\ 5 & \frac{12}{7} \end{array}$

Step 5

Graph 2x^2-12x+28y=38

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