Algebra Examples

$(- 4, 2)$ , $(6, 10)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 4, 2)$ as the vertex $(h, k)$ and $(6, 10)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$10 = a {(6 - (- 4))}^{2} + 2$

Step 2

Using $10 = a {(6 - (- 4))}^{2} + 2$ to solve for $a$ , $a = \frac{2}{25}$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $a {(6 - (- 4))}^{2} + 2 = 10$ .

$a {(6 - (- 4))}^{2} + 2 = 10$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Multiply $- 1$ by $- 4$ .

$a {(6 + 4)}^{2} + 2 = 10$

Step 2.2.2

Add $6$ and $4$ .

$a \cdot 10^{2} + 2 = 10$

Step 2.2.3

Raise $10$ to the power of $2$ .

$a \cdot 100 + 2 = 10$

Step 2.2.4

Move $100$ to the left of $a$ .

$100 a + 2 = 10$

Step 2.3

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

Tap for more steps...

Step 2.3.1

Subtract $2$ from both sides of the equation.

$100 a = 10 - 2$

Step 2.3.2

Subtract $2$ from $10$ .

$100 a = 8$

Step 2.4

Divide each term in $100 a = 8$ by $100$ and simplify.

Tap for more steps...

Step 2.4.1

Divide each term in $100 a = 8$ by $100$ .

$\frac{100 a}{100} = \frac{8}{100}$

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $100$ .

Tap for more steps...

Step 2.4.2.1.1

Cancel the common factor.

$\frac{100 a}{100} = \frac{8}{100}$

Step 2.4.2.1.2

Divide $a$ by $1$ .

$a = \frac{8}{100}$

Step 2.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.3.1

Cancel the common factor of $8$ and $100$ .

Tap for more steps...

Step 2.4.3.1.1

Factor $4$ out of $8$ .

$a = \frac{4 (2)}{100}$

Step 2.4.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.4.3.1.2.1

Factor $4$ out of $100$ .

$a = \frac{4 \cdot 2}{4 \cdot 25}$

Step 2.4.3.1.2.2

Cancel the common factor.

$a = \frac{4 \cdot 2}{4 \cdot 25}$

Step 2.4.3.1.2.3

Rewrite the expression.

$a = \frac{2}{25}$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 4, 2)$ and $a = \frac{2}{25}$ is $y = (\frac{2}{25}) {(x - (- 4))}^{2} + 2$ .

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

Step 4

Solve $y = (\frac{2}{25}) {(x - (- 4))}^{2} + 2$ for $y$ .

Tap for more steps...

Step 4.1

Remove parentheses.

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

Step 4.2

Multiply $\frac{2}{25}$ by ${(x - (- 4))}^{2}$ .

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

Step 4.3

Remove parentheses.

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

Step 4.4

Simplify $(\frac{2}{25}) {(x - (- 4))}^{2} + 2$ .

Tap for more steps...

Step 4.4.1

Simplify each term.

Tap for more steps...

Step 4.4.1.1

Multiply $- 1$ by $- 4$ .

$y = \frac{2}{25} {(x + 4)}^{2} + 2$

Step 4.4.1.2

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$y = \frac{2}{25} ((x + 4) (x + 4)) + 2$

Step 4.4.1.3

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Tap for more steps...

Step 4.4.1.3.1

Apply the distributive property.

$y = \frac{2}{25} (x (x + 4) + 4 (x + 4)) + 2$

Step 4.4.1.3.2

Apply the distributive property.

$y = \frac{2}{25} (x \cdot x + x \cdot 4 + 4 (x + 4)) + 2$

Step 4.4.1.3.3

Apply the distributive property.

$y = \frac{2}{25} (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) + 2$

Step 4.4.1.4

Simplify and combine like terms.

Tap for more steps...

Step 4.4.1.4.1

Simplify each term.

Tap for more steps...

Step 4.4.1.4.1.1

Multiply $x$ by $x$ .

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

Step 4.4.1.4.1.2

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

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

Step 4.4.1.4.1.3

Multiply $4$ by $4$ .

$y = \frac{2}{25} (x^{2} + 4 x + 4 x + 16) + 2$

Step 4.4.1.4.2

Add $4 x$ and $4 x$ .

$y = \frac{2}{25} (x^{2} + 8 x + 16) + 2$

Step 4.4.1.5

Apply the distributive property.

$y = \frac{2}{25} x^{2} + \frac{2}{25} (8 x) + \frac{2}{25} \cdot 16 + 2$

Step 4.4.1.6

Simplify.

Tap for more steps...

Step 4.4.1.6.1

Combine $\frac{2}{25}$ and $x^{2}$ .

$y = \frac{2 x^{2}}{25} + \frac{2}{25} (8 x) + \frac{2}{25} \cdot 16 + 2$

Step 4.4.1.6.2

Multiply $\frac{2}{25} (8 x)$ .

Tap for more steps...

Step 4.4.1.6.2.1

Combine $8$ and $\frac{2}{25}$ .

$y = \frac{2 x^{2}}{25} + \frac{8 \cdot 2}{25} x + \frac{2}{25} \cdot 16 + 2$

Step 4.4.1.6.2.2

Multiply $8$ by $2$ .

$y = \frac{2 x^{2}}{25} + \frac{16}{25} x + \frac{2}{25} \cdot 16 + 2$

Step 4.4.1.6.2.3

Combine $\frac{16}{25}$ and $x$ .

$y = \frac{2 x^{2}}{25} + \frac{16 x}{25} + \frac{2}{25} \cdot 16 + 2$

Step 4.4.1.6.3

Multiply $\frac{2}{25} \cdot 16$ .

Tap for more steps...

Step 4.4.1.6.3.1

Combine $\frac{2}{25}$ and $16$ .

$y = \frac{2 x^{2}}{25} + \frac{16 x}{25} + \frac{2 \cdot 16}{25} + 2$

Step 4.4.1.6.3.2

Multiply $2$ by $16$ .

$y = \frac{2 x^{2}}{25} + \frac{16 x}{25} + \frac{32}{25} + 2$

Step 4.4.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y = \frac{2 x^{2}}{25} + \frac{16 x}{25} + \frac{32}{25} + 2 \cdot \frac{25}{25}$

Step 4.4.3

Combine $2$ and $\frac{25}{25}$ .

$y = \frac{2 x^{2}}{25} + \frac{16 x}{25} + \frac{32}{25} + \frac{2 \cdot 25}{25}$

Step 4.4.4

Combine the numerators over the common denominator.

$y = \frac{2 x^{2}}{25} + \frac{16 x}{25} + \frac{32 + 2 \cdot 25}{25}$

Step 4.4.5

Simplify the numerator.

Tap for more steps...

Step 4.4.5.1

Multiply $2$ by $25$ .

$y = \frac{2 x^{2}}{25} + \frac{16 x}{25} + \frac{32 + 50}{25}$

Step 4.4.5.2

Add $32$ and $50$ .

$y = \frac{2 x^{2}}{25} + \frac{16 x}{25} + \frac{82}{25}$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = \frac{2}{25} x^{2} + \frac{16}{25} x + \frac{82}{25}$

Vertex Form: $y = (\frac{2}{25}) {(x - (- 4))}^{2} + 2$

Step 6

Simplify the standard form.

Standard Form: $y = \frac{2}{25} x^{2} + \frac{16}{25} x + \frac{82}{25}$

Vertex Form: $y = \frac{2}{25} {(x + 4)}^{2} + 2$

Step 7