Algebra Examples

$(- 5, 0)$ , $(- 3, 1)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 3, 1)$ as the vertex $(h, k)$ and $(- 5, 0)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$0 = a {(- 5 - (- 3))}^{2} + 1$

Step 2

Using $0 = a {(- 5 - (- 3))}^{2} + 1$ to solve for $a$ , $a = - \frac{1}{4}$ .

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Step 2.1

Rewrite the equation as $a {(- 5 - (- 3))}^{2} + 1 = 0$ .

$a {(- 5 - (- 3))}^{2} + 1 = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $- 1$ by $- 3$ .

$a {(- 5 + 3)}^{2} + 1 = 0$

Step 2.2.2

Add $- 5$ and $3$ .

$a {(- 2)}^{2} + 1 = 0$

Step 2.2.3

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

$a \cdot 4 + 1 = 0$

Step 2.2.4

Move $4$ to the left of $a$ .

$4 a + 1 = 0$

Step 2.3

Subtract $1$ from both sides of the equation.

$4 a = - 1$

Step 2.4

Divide each term in $4 a = - 1$ by $4$ and simplify.

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Step 2.4.1

Divide each term in $4 a = - 1$ by $4$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Divide $a$ by $1$ .

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Move the negative in front of the fraction.

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

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 3, 1)$ and $a = - \frac{1}{4}$ is $y = (- \frac{1}{4}) {(x - (- 3))}^{2} + 1$ .

$y = (- \frac{1}{4}) {(x - (- 3))}^{2} + 1$

Step 4

Solve $y = (- \frac{1}{4}) {(x - (- 3))}^{2} + 1$ for $y$ .

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Step 4.1

Remove parentheses.

$y = (- \frac{1}{4}) {(x - (- 3))}^{2} + 1$

Step 4.2

Multiply $- \frac{1}{4}$ by ${(x - (- 3))}^{2}$ .

$y = - \frac{1}{4} \cdot {(x - (- 3))}^{2} + 1$

Step 4.3

Remove parentheses.

$y = (- \frac{1}{4}) {(x - (- 3))}^{2} + 1$

Step 4.4

Simplify $(- \frac{1}{4}) {(x - (- 3))}^{2} + 1$ .

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Multiply $- 1$ by $- 3$ .

$y = - \frac{1}{4} {(x + 3)}^{2} + 1$

Step 4.4.1.2

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$y = - \frac{1}{4} ((x + 3) (x + 3)) + 1$

Step 4.4.1.3

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

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Step 4.4.1.3.1

Apply the distributive property.

$y = - \frac{1}{4} (x (x + 3) + 3 (x + 3)) + 1$

Step 4.4.1.3.2

Apply the distributive property.

$y = - \frac{1}{4} (x \cdot x + x \cdot 3 + 3 (x + 3)) + 1$

Step 4.4.1.3.3

Apply the distributive property.

$y = - \frac{1}{4} (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + 1$

Step 4.4.1.4

Simplify and combine like terms.

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Step 4.4.1.4.1

Simplify each term.

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Step 4.4.1.4.1.1

Multiply $x$ by $x$ .

$y = - \frac{1}{4} (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) + 1$

Step 4.4.1.4.1.2

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

$y = - \frac{1}{4} (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) + 1$

Step 4.4.1.4.1.3

Multiply $3$ by $3$ .

$y = - \frac{1}{4} (x^{2} + 3 x + 3 x + 9) + 1$

Step 4.4.1.4.2

Add $3 x$ and $3 x$ .

$y = - \frac{1}{4} (x^{2} + 6 x + 9) + 1$

Step 4.4.1.5

Apply the distributive property.

$y = - \frac{1}{4} x^{2} - \frac{1}{4} (6 x) - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6

Simplify.

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Step 4.4.1.6.1

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

$y = - \frac{x^{2}}{4} - \frac{1}{4} (6 x) - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6.2

Cancel the common factor of $2$ .

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Step 4.4.1.6.2.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$y = - \frac{x^{2}}{4} + \frac{- 1}{4} (6 x) - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6.2.2

Factor $2$ out of $4$ .

$y = - \frac{x^{2}}{4} + \frac{- 1}{2 (2)} (6 x) - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6.2.3

Factor $2$ out of $6 x$ .

$y = - \frac{x^{2}}{4} + \frac{- 1}{2 (2)} (2 (3 x)) - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6.2.4

Cancel the common factor.

$y = - \frac{x^{2}}{4} + \frac{- 1}{2 \cdot 2} (2 (3 x)) - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6.2.5

Rewrite the expression.

$y = - \frac{x^{2}}{4} + \frac{- 1}{2} (3 x) - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6.3

Combine $3$ and $\frac{- 1}{2}$ .

$y = - \frac{x^{2}}{4} + \frac{3 \cdot - 1}{2} x - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6.4

Multiply $3$ by $- 1$ .

$y = - \frac{x^{2}}{4} + \frac{- 3}{2} x - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6.5

Combine $\frac{- 3}{2}$ and $x$ .

$y = - \frac{x^{2}}{4} + \frac{- 3 x}{2} - \frac{1}{4} \cdot 9 + 1$

Step 4.4.1.6.6

Multiply $- \frac{1}{4} \cdot 9$ .

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Step 4.4.1.6.6.1

Multiply $9$ by $- 1$ .

$y = - \frac{x^{2}}{4} + \frac{- 3 x}{2} - 9 (\frac{1}{4}) + 1$

Step 4.4.1.6.6.2

Combine $- 9$ and $\frac{1}{4}$ .

$y = - \frac{x^{2}}{4} + \frac{- 3 x}{2} + \frac{- 9}{4} + 1$

Step 4.4.1.7

Simplify each term.

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Step 4.4.1.7.1

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{4} - \frac{3 x}{2} + \frac{- 9}{4} + 1$

Step 4.4.1.7.2

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{4} - \frac{3 x}{2} - \frac{9}{4} + 1$

Step 4.4.2

Simplify the expression.

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Step 4.4.2.1

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

$y = - \frac{x^{2}}{4} - \frac{3 x}{2} - \frac{9}{4} + \frac{4}{4}$

Step 4.4.2.2

Combine the numerators over the common denominator.

$y = - \frac{x^{2}}{4} - \frac{3 x}{2} + \frac{- 9 + 4}{4}$

Step 4.4.2.3

Add $- 9$ and $4$ .

$y = - \frac{x^{2}}{4} - \frac{3 x}{2} + \frac{- 5}{4}$

Step 4.4.2.4

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{4} - \frac{3 x}{2} - \frac{5}{4}$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - \frac{1}{4} x^{2} - \frac{3}{2} x - \frac{5}{4}$

Vertex Form: $y = (- \frac{1}{4}) {(x - (- 3))}^{2} + 1$

Step 6

Simplify the standard form.

Standard Form: $y = - \frac{1}{4} x^{2} - \frac{3}{2} x - \frac{5}{4}$

Vertex Form: $y = - \frac{1}{4} {(x + 3)}^{2} + 1$

Step 7