Find the Parabola Through (1,12) with Vertex (10,-4) (1,12) , (10,-4)

$(1, 12)$ , $(10, - 4)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(10, - 4)$ as the vertex $(h, k)$ and $(1, 12)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$12 = a {(1 - (10))}^{2} - 4$

Step 2

Using $12 = a {(1 - (10))}^{2} - 4$ to solve for $a$ , $a = \frac{16}{81}$ .

Rewrite the equation as $a {(1 - (10))}^{2} - 4 = 12$ .

$a {(1 - (10))}^{2} - 4 = 12$

Simplify each term.

Multiply $- 1$ by $10$ .

$a {(1 - 10)}^{2} - 4 = 12$

Subtract $10$ from $1$ .

$a {(- 9)}^{2} - 4 = 12$

Raise $- 9$ to the power of $2$ .

$a \cdot 81 - 4 = 12$

Move $81$ to the left of $a$ .

$81 a - 4 = 12$

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

Add $4$ to both sides of the equation.

$81 a = 12 + 4$

Add $12$ and $4$ .

$81 a = 16$

Divide each term in $81 a = 16$ by $81$ and simplify.

Divide each term in $81 a = 16$ by $81$ .

$\frac{81 a}{81} = \frac{16}{81}$

Simplify the left side.

Cancel the common factor of $81$ .

Cancel the common factor.

$\frac{81 a}{81} = \frac{16}{81}$

Divide $a$ by $1$ .

$a = \frac{16}{81}$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(10, - 4)$ and $a = \frac{16}{81}$ is $y = (\frac{16}{81}) {(x - (10))}^{2} - 4$ .

$y = (\frac{16}{81}) {(x - (10))}^{2} - 4$

Step 4

Solve $y = (\frac{16}{81}) {(x - (10))}^{2} - 4$ for $y$ .

Remove parentheses.

$y = (\frac{16}{81}) {(x - (10))}^{2} - 4$

Multiply $\frac{16}{81}$ by ${(x - (10))}^{2}$ .

$y = \frac{16}{81} \cdot {(x - (10))}^{2} - 4$

Remove parentheses.

$y = (\frac{16}{81}) {(x - (10))}^{2} - 4$

Simplify $(\frac{16}{81}) {(x - (10))}^{2} - 4$ .

Simplify each term.

Multiply $- 1$ by $10$ .

$y = \frac{16}{81} {(x - 10)}^{2} - 4$

Rewrite ${(x - 10)}^{2}$ as $(x - 10) (x - 10)$ .

$y = \frac{16}{81} ((x - 10) (x - 10)) - 4$

Expand $(x - 10) (x - 10)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{16}{81} (x (x - 10) - 10 (x - 10)) - 4$

Apply the distributive property.

$y = \frac{16}{81} (x \cdot x + x \cdot - 10 - 10 (x - 10)) - 4$

Apply the distributive property.

$y = \frac{16}{81} (x \cdot x + x \cdot - 10 - 10 x - 10 \cdot - 10) - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = \frac{16}{81} (x^{2} + x \cdot - 10 - 10 x - 10 \cdot - 10) - 4$

Move $- 10$ to the left of $x$ .

$y = \frac{16}{81} (x^{2} - 10 \cdot x - 10 x - 10 \cdot - 10) - 4$

Multiply $- 10$ by $- 10$ .

$y = \frac{16}{81} (x^{2} - 10 x - 10 x + 100) - 4$

Subtract $10 x$ from $- 10 x$ .

$y = \frac{16}{81} (x^{2} - 20 x + 100) - 4$

Apply the distributive property.

$y = \frac{16}{81} x^{2} + \frac{16}{81} (- 20 x) + \frac{16}{81} \cdot 100 - 4$

Simplify.

Combine $\frac{16}{81}$ and $x^{2}$ .

$y = \frac{16 x^{2}}{81} + \frac{16}{81} (- 20 x) + \frac{16}{81} \cdot 100 - 4$

Multiply $\frac{16}{81} (- 20 x)$ .

Combine $- 20$ and $\frac{16}{81}$ .

$y = \frac{16 x^{2}}{81} + \frac{- 20 \cdot 16}{81} x + \frac{16}{81} \cdot 100 - 4$

Multiply $- 20$ by $16$ .

$y = \frac{16 x^{2}}{81} + \frac{- 320}{81} x + \frac{16}{81} \cdot 100 - 4$

Combine $\frac{- 320}{81}$ and $x$ .

$y = \frac{16 x^{2}}{81} + \frac{- 320 x}{81} + \frac{16}{81} \cdot 100 - 4$

Multiply $\frac{16}{81} \cdot 100$ .

Combine $\frac{16}{81}$ and $100$ .

$y = \frac{16 x^{2}}{81} + \frac{- 320 x}{81} + \frac{16 \cdot 100}{81} - 4$

Multiply $16$ by $100$ .

$y = \frac{16 x^{2}}{81} + \frac{- 320 x}{81} + \frac{1600}{81} - 4$

Move the negative in front of the fraction.

$y = \frac{16 x^{2}}{81} - \frac{320 x}{81} + \frac{1600}{81} - 4$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

$y = \frac{16 x^{2}}{81} - \frac{320 x}{81} + \frac{1600}{81} - 4 \cdot \frac{81}{81}$

Combine $- 4$ and $\frac{81}{81}$ .

$y = \frac{16 x^{2}}{81} - \frac{320 x}{81} + \frac{1600}{81} + \frac{- 4 \cdot 81}{81}$

Combine the numerators over the common denominator.

$y = \frac{16 x^{2}}{81} - \frac{320 x}{81} + \frac{1600 - 4 \cdot 81}{81}$

Simplify the numerator.

Multiply $- 4$ by $81$ .

$y = \frac{16 x^{2}}{81} - \frac{320 x}{81} + \frac{1600 - 324}{81}$

Subtract $324$ from $1600$ .

$y = \frac{16 x^{2}}{81} - \frac{320 x}{81} + \frac{1276}{81}$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = \frac{16}{81} x^{2} - \frac{320}{81} x + \frac{1276}{81}$

Vertex Form: $y = (\frac{16}{81}) {(x - (10))}^{2} - 4$

Step 6

Simplify the standard form.

Standard Form: $y = \frac{16}{81} x^{2} - \frac{320}{81} x + \frac{1276}{81}$

Vertex Form: $y = \frac{16}{81} {(x - 10)}^{2} - 4$

Step 7

Find the Parabola Through (1,12) with Vertex (10,-4) (1,12) , (10,-4)

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