Graph y=-3/8(x+5)(x-3)

$y = - \frac{3}{8} (x + 5) (x - 3)$

Solve for $y$ .

Remove parentheses.

$y = - \frac{3}{8} \cdot ((x + 5) (x - 3))$

Remove parentheses.

$y = - \frac{3}{8} \cdot ((x + 5) (x - 3))$

Simplify $- \frac{3}{8} \cdot ((x + 5) (x - 3))$ .

Expand $(x + 5) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$y = - \frac{3}{8} \cdot (x (x - 3) + 5 (x - 3))$

Apply the distributive property.

$y = - \frac{3}{8} \cdot (x \cdot x + x \cdot - 3 + 5 (x - 3))$

Apply the distributive property.

$y = - \frac{3}{8} \cdot (x \cdot x + x \cdot - 3 + 5 x + 5 \cdot - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = - \frac{3}{8} \cdot (x^{2} + x \cdot - 3 + 5 x + 5 \cdot - 3)$

Move $- 3$ to the left of $x$ .

$y = - \frac{3}{8} \cdot (x^{2} - 3 \cdot x + 5 x + 5 \cdot - 3)$

Multiply $5$ by $- 3$ .

$y = - \frac{3}{8} \cdot (x^{2} - 3 x + 5 x - 15)$

Add $- 3 x$ and $5 x$ .

$y = - \frac{3}{8} \cdot (x^{2} + 2 x - 15)$

Apply the distributive property.

$y = - \frac{3}{8} x^{2} - \frac{3}{8} (2 x) - \frac{3}{8} \cdot - 15$

Simplify.

Combine $x^{2}$ and $\frac{3}{8}$ .

$y = - \frac{x^{2} \cdot 3}{8} - \frac{3}{8} (2 x) - \frac{3}{8} \cdot - 15$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{8}$ into the numerator.

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3}{8} (2 x) - \frac{3}{8} \cdot - 15$

Factor $2$ out of $8$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3}{2 (4)} (2 x) - \frac{3}{8} \cdot - 15$

Factor $2$ out of $2 x$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3}{2 (4)} (2 (x)) - \frac{3}{8} \cdot - 15$

Cancel the common factor.

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3}{2 \cdot 4} (2 x) - \frac{3}{8} \cdot - 15$

Rewrite the expression.

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3}{4} x - \frac{3}{8} \cdot - 15$

Combine $\frac{- 3}{4}$ and $x$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3 x}{4} - \frac{3}{8} \cdot - 15$

Multiply $- \frac{3}{8} \cdot - 15$ .

Multiply $- 15$ by $- 1$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3 x}{4} + 15 (\frac{3}{8})$

Combine $15$ and $\frac{3}{8}$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3 x}{4} + \frac{15 \cdot 3}{8}$

Multiply $15$ by $3$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3 x}{4} + \frac{45}{8}$

Simplify each term.

Move $3$ to the left of $x^{2}$ .

$y = - \frac{3 \cdot x^{2}}{8} + \frac{- 3 x}{4} + \frac{45}{8}$

Move the negative in front of the fraction.

$y = - \frac{3 x^{2}}{8} - \frac{3 x}{4} + \frac{45}{8}$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- \frac{3 x^{2}}{8} - \frac{3 x}{4} + \frac{45}{8}$ .

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

$a = - \frac{3}{8}$

$b = - \frac{3}{4}$

$c = \frac{45}{8}$

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

$d = \frac{- \frac{3}{4}}{2 (- \frac{3}{8})}$

Simplify the right side.

Dividing two negative values results in a positive value.

$d = \frac{\frac{3}{4}}{2 (\frac{3}{8})}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{4} \cdot \frac{1}{2 (\frac{3}{8})}$

Combine $2$ and $\frac{3}{8}$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 \cdot 3}{8}}$

Multiply $2$ by $3$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{6}{8}}$

Cancel the common factor of $6$ and $8$ .

Factor $2$ out of $6$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 (3)}{8}}$

Cancel the common factors.

Factor $2$ out of $8$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 \cdot 3}{2 \cdot 4}}$

Cancel the common factor.

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 \cdot 3}{2 \cdot 4}}$

Rewrite the expression.

$d = \frac{3}{4} \cdot \frac{1}{\frac{3}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{4} (1 (\frac{4}{3}))$

Multiply $\frac{4}{3}$ by $1$ .

$d = \frac{3}{4} \cdot \frac{4}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$d = \frac{3}{4} \cdot \frac{4}{3}$

Rewrite the expression.

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

$e = \frac{45}{8} - \frac{{(- \frac{3}{4})}^{2}}{4 (- \frac{3}{8})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{3}{4}$ .

$e = \frac{45}{8} - \frac{{(- 1)}^{2} {(\frac{3}{4})}^{2}}{4 (- \frac{3}{8})}$

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

$e = \frac{45}{8} - \frac{1 {(\frac{3}{4})}^{2}}{4 (- \frac{3}{8})}$

Apply the product rule to $\frac{3}{4}$ .

$e = \frac{45}{8} - \frac{1 \frac{3^{2}}{4^{2}}}{4 (- \frac{3}{8})}$

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

$e = \frac{45}{8} - \frac{1 \frac{9}{4^{2}}}{4 (- \frac{3}{8})}$

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

$e = \frac{45}{8} - \frac{1 (\frac{9}{16})}{4 (- \frac{3}{8})}$

Multiply $\frac{9}{16}$ by $1$ .

$e = \frac{45}{8} - \frac{\frac{9}{16}}{4 (- \frac{3}{8})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = \frac{45}{8} - \frac{\frac{9}{16}}{- 4 (\frac{3}{8})}$

Combine $- 4$ and $\frac{3}{8}$ .

$e = \frac{45}{8} - \frac{\frac{9}{16}}{\frac{- 4 \cdot 3}{8}}$

Multiply $- 4$ by $3$ .

$e = \frac{45}{8} - \frac{\frac{9}{16}}{\frac{- 12}{8}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 12$ and $8$ .

Factor $4$ out of $- 12$ .

$e = \frac{45}{8} - \frac{\frac{9}{16}}{\frac{4 (- 3)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$e = \frac{45}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot - 3}{4 \cdot 2}}$

Cancel the common factor.

$e = \frac{45}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot - 3}{4 \cdot 2}}$

Rewrite the expression.

$e = \frac{45}{8} - \frac{\frac{9}{16}}{\frac{- 3}{2}}$

Move the negative in front of the fraction.

$e = \frac{45}{8} - \frac{\frac{9}{16}}{- \frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{45}{8} - (\frac{9}{16} (- \frac{2}{3}))$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$e = \frac{45}{8} - (\frac{9}{16} \cdot \frac{- 2}{3})$

Factor $3$ out of $9$ .

$e = \frac{45}{8} - (\frac{3 (3)}{16} \cdot \frac{- 2}{3})$

Cancel the common factor.

$e = \frac{45}{8} - (\frac{3 \cdot 3}{16} \cdot \frac{- 2}{3})$

Rewrite the expression.

$e = \frac{45}{8} - (\frac{3}{16} \cdot - 2)$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$e = \frac{45}{8} - (\frac{3}{2 (8)} \cdot - 2)$

Factor $2$ out of $- 2$ .

$e = \frac{45}{8} - (\frac{3}{2 \cdot 8} \cdot (2 \cdot - 1))$

Cancel the common factor.

$e = \frac{45}{8} - (\frac{3}{2 \cdot 8} \cdot (2 \cdot - 1))$

Rewrite the expression.

$e = \frac{45}{8} - (\frac{3}{8} \cdot - 1)$

Combine $\frac{3}{8}$ and $- 1$ .

$e = \frac{45}{8} - \frac{3 \cdot - 1}{8}$

Multiply $3$ by $- 1$ .

$e = \frac{45}{8} - \frac{- 3}{8}$

Move the negative in front of the fraction.

$e = \frac{45}{8} - - \frac{3}{8}$

Multiply $- - \frac{3}{8}$ .

Multiply $- 1$ by $- 1$ .

$e = \frac{45}{8} + 1 (\frac{3}{8})$

Multiply $\frac{3}{8}$ by $1$ .

$e = \frac{45}{8} + \frac{3}{8}$

Combine the numerators over the common denominator.

$e = \frac{45 + 3}{8}$

Add $45$ and $3$ .

$e = \frac{48}{8}$

Divide $48$ by $8$ .

$e = 6$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{3}{8} {(x + 1)}^{2} + 6$ .

$- \frac{3}{8} {(x + 1)}^{2} + 6$

Set $y$ equal to the new right side.

$y = - \frac{3}{8} \cdot {(x + 1)}^{2} + 6$

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

$a = - \frac{3}{8}$

$h = - 1$

$k = 6$

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

Opens Down

Find the vertex $(h, k)$ .

$(- 1, 6)$

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.

$\frac{1}{4 \cdot (- \frac{3}{8})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{3}{8})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{3}{8})}$

Combine $4$ and $\frac{3}{8}$ .

$- \frac{1}{\frac{4 \cdot 3}{8}}$

Multiply $4$ by $3$ .

$- \frac{1}{\frac{12}{8}}$

Cancel the common factor of $12$ and $8$ .

Factor $4$ out of $12$ .

$- \frac{1}{\frac{4 (3)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$- \frac{1}{\frac{4 \cdot 3}{4 \cdot 2}}$

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 3}{4 \cdot 2}}$

Rewrite the expression.

$- \frac{1}{\frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{2}{3}))$

Multiply $\frac{2}{3}$ by $1$ .

$- \frac{2}{3}$

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.

$(- 1, \frac{16}{3})$

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

$x = - 1$

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.

$y = \frac{20}{3}$

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

Direction: Opens Down

Vertex: $(- 1, 6)$

Focus: $(- 1, \frac{16}{3})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{20}{3}$

Direction: Opens Down

Vertex: $(- 1, 6)$

Focus: $(- 1, \frac{16}{3})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{20}{3}$

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.

$f (- 2) = - \frac{3 {(- 2)}^{2}}{8} - \frac{3 (- 2)}{4} + \frac{45}{8}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 3 {(- 2)}^{2} + 45}{8} + \frac{- 3 \cdot - 2}{4}$

Simplify each term.

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

$f (- 2) = \frac{- 3 \cdot 4 + 45}{8} + \frac{- 3 \cdot - 2}{4}$

Multiply $- 3$ by $4$ .

$f (- 2) = \frac{- 12 + 45}{8} + \frac{- 3 \cdot - 2}{4}$

Reduce the expression by cancelling the common factors.

Add $- 12$ and $45$ .

$f (- 2) = \frac{33}{8} + \frac{- 3 \cdot - 2}{4}$

Multiply $- 3$ by $- 2$ .

$f (- 2) = \frac{33}{8} + \frac{6}{4}$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$f (- 2) = \frac{33}{8} + \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (- 2) = \frac{33}{8} + \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$f (- 2) = \frac{33}{8} + \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$f (- 2) = \frac{33}{8} + \frac{3}{2}$

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- 2) = \frac{33}{8} + \frac{3}{2} \cdot \frac{4}{4}$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{2}$ by $\frac{4}{4}$ .

$f (- 2) = \frac{33}{8} + \frac{3 \cdot 4}{2 \cdot 4}$

Multiply $2$ by $4$ .

$f (- 2) = \frac{33}{8} + \frac{3 \cdot 4}{8}$

Combine the numerators over the common denominator.

$f (- 2) = \frac{33 + 3 \cdot 4}{8}$

Simplify the numerator.

Multiply $3$ by $4$ .

$f (- 2) = \frac{33 + 12}{8}$

Add $33$ and $12$ .

$f (- 2) = \frac{45}{8}$

The final answer is $\frac{45}{8}$ .

$\frac{45}{8}$

The $y$ value at $x = - 2$ is $\frac{45}{8}$ .

$y = \frac{45}{8}$

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

$f (- 3) = - \frac{3 {(- 3)}^{2}}{8} - \frac{3 (- 3)}{4} + \frac{45}{8}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 3 {(- 3)}^{2} + 45}{8} + \frac{- 3 \cdot - 3}{4}$

Simplify each term.

Multiply $- 3$ by ${(- 3)}^{2}$ by adding the exponents.

Multiply $- 3$ by ${(- 3)}^{2}$ .

Raise $- 3$ to the power of $1$ .

$f (- 3) = \frac{(- 3) {(- 3)}^{2} + 45}{8} + \frac{- 3 \cdot - 3}{4}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 3) = \frac{{(- 3)}^{1 + 2} + 45}{8} + \frac{- 3 \cdot - 3}{4}$

Add $1$ and $2$ .

$f (- 3) = \frac{{(- 3)}^{3} + 45}{8} + \frac{- 3 \cdot - 3}{4}$

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

$f (- 3) = \frac{- 27 + 45}{8} + \frac{- 3 \cdot - 3}{4}$

Simplify terms.

Add $- 27$ and $45$ .

$f (- 3) = \frac{18}{8} + \frac{- 3 \cdot - 3}{4}$

Multiply $- 3$ by $- 3$ .

$f (- 3) = \frac{18}{8} + \frac{9}{4}$

Cancel the common factor of $18$ and $8$ .

Factor $2$ out of $18$ .

$f (- 3) = \frac{2 (9)}{8} + \frac{9}{4}$

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$f (- 3) = \frac{9}{4} + \frac{9}{4}$

Combine the numerators over the common denominator.

$f (- 3) = \frac{9 + 9}{4}$

Add $9$ and $9$ .

$f (- 3) = \frac{18}{4}$

Cancel the common factor of $18$ and $4$ .

Factor $2$ out of $18$ .

$f (- 3) = \frac{2 (9)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (- 3) = \frac{2 \cdot 9}{2 \cdot 2}$

Cancel the common factor.

$f (- 3) = \frac{2 \cdot 9}{2 \cdot 2}$

Rewrite the expression.

$f (- 3) = \frac{9}{2}$

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

The $y$ value at $x = - 3$ is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \frac{3 {(0)}^{2}}{8} - \frac{3 (0)}{4} + \frac{45}{8}$

Simplify the result.

Combine the numerators over the common denominator.

$f (0) = \frac{- 3 {(0)}^{2} + 45}{8} + \frac{- 3 \cdot 0}{4}$

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{- 3 \cdot 0 + 45}{8} + \frac{- 3 \cdot 0}{4}$

Multiply $- 3$ by $0$ .

$f (0) = \frac{0 + 45}{8} + \frac{- 3 \cdot 0}{4}$

Simplify the expression.

Add $0$ and $45$ .

$f (0) = \frac{45}{8} + \frac{- 3 \cdot 0}{4}$

Multiply $- 3$ by $0$ .

$f (0) = \frac{45}{8} + \frac{0}{4}$

Divide $0$ by $4$ .

$f (0) = \frac{45}{8} + 0$

Add $\frac{45}{8}$ and $0$ .

$f (0) = \frac{45}{8}$

The final answer is $\frac{45}{8}$ .

$\frac{45}{8}$

The $y$ value at $x = 0$ is $\frac{45}{8}$ .

$y = \frac{45}{8}$

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

$f (1) = - \frac{3 {(1)}^{2}}{8} - \frac{3 (1)}{4} + \frac{45}{8}$

Simplify the result.

Combine the numerators over the common denominator.

$f (1) = \frac{- 3 {(1)}^{2} + 45}{8} + \frac{- 3 \cdot 1}{4}$

Simplify each term.

One to any power is one.

$f (1) = \frac{- 3 \cdot 1 + 45}{8} + \frac{- 3 \cdot 1}{4}$

Multiply $- 3$ by $1$ .

$f (1) = \frac{- 3 + 45}{8} + \frac{- 3 \cdot 1}{4}$

Simplify the expression.

Add $- 3$ and $45$ .

$f (1) = \frac{42}{8} + \frac{- 3 \cdot 1}{4}$

Multiply $- 3$ by $1$ .

$f (1) = \frac{42}{8} + \frac{- 3}{4}$

Simplify each term.

Cancel the common factor of $42$ and $8$ .

Factor $2$ out of $42$ .

$f (1) = \frac{2 (21)}{8} + \frac{- 3}{4}$

Cancel the common factors.

Factor $2$ out of $8$ .

$f (1) = \frac{2 \cdot 21}{2 \cdot 4} + \frac{- 3}{4}$

Cancel the common factor.

$f (1) = \frac{2 \cdot 21}{2 \cdot 4} + \frac{- 3}{4}$

Rewrite the expression.

$f (1) = \frac{21}{4} + \frac{- 3}{4}$

Move the negative in front of the fraction.

$f (1) = \frac{21}{4} - \frac{3}{4}$

Simplify terms.

Combine the numerators over the common denominator.

$f (1) = \frac{21 - 3}{4}$

Subtract $3$ from $21$ .

$f (1) = \frac{18}{4}$

Cancel the common factor of $18$ and $4$ .

Factor $2$ out of $18$ .

$f (1) = \frac{2 (9)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$f (1) = \frac{9}{2}$

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

The $y$ value at $x = 1$ is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & \frac{9}{2} \\ - 2 & \frac{45}{8} \\ - 1 & 6 \\ 0 & \frac{45}{8} \\ 1 & \frac{9}{2} \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 1, 6)$

Focus: $(- 1, \frac{16}{3})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{20}{3}$

$\begin{array}{c} x & y \\ - 3 & \frac{9}{2} \\ - 2 & \frac{45}{8} \\ - 1 & 6 \\ 0 & \frac{45}{8} \\ 1 & \frac{9}{2} \end{array}$

Graph y=-3/8(x+5)(x-3)

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