Find the Parabola Through (6,1) with Vertex (5,4) (5,4) , (6,1)

$(5, 4)$ , $(6, 1)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(5, 4)$ as the vertex $(h, k)$ and $(6, 1)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$1 = a {(6 - (5))}^{2} + 4$

Step 2

Using $1 = a {(6 - (5))}^{2} + 4$ to solve for $a$ , $a = - 3$ .

Rewrite the equation as $a {(6 - (5))}^{2} + 4 = 1$ .

$a {(6 - (5))}^{2} + 4 = 1$

Simplify each term.

Multiply $- 1$ by $5$ .

$a {(6 - 5)}^{2} + 4 = 1$

Subtract $5$ from $6$ .

$a \cdot 1^{2} + 4 = 1$

One to any power is one.

$a \cdot 1 + 4 = 1$

Multiply $a$ by $1$ .

$a + 4 = 1$

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

Subtract $4$ from both sides of the equation.

$a = 1 - 4$

Subtract $4$ from $1$ .

$a = - 3$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(5, 4)$ and $a = - 3$ is $y = (- 3) {(x - (5))}^{2} + 4$ .

$y = (- 3) {(x - (5))}^{2} + 4$

Step 4

Solve $y = (- 3) {(x - (5))}^{2} + 4$ for $y$ .

Remove parentheses.

$y = (- 3) {(x - (5))}^{2} + 4$

Multiply $- 3$ by ${(x - (5))}^{2}$ .

$y = - 3 {(x - (5))}^{2} + 4$

Remove parentheses.

$y = (- 3) {(x - (5))}^{2} + 4$

Simplify $(- 3) {(x - (5))}^{2} + 4$ .

Simplify each term.

Multiply $- 1$ by $5$ .

$y = - 3 {(x - 5)}^{2} + 4$

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$y = - 3 ((x - 5) (x - 5)) + 4$

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

Apply the distributive property.

$y = - 3 (x (x - 5) - 5 (x - 5)) + 4$

Apply the distributive property.

$y = - 3 (x \cdot x + x \cdot - 5 - 5 (x - 5)) + 4$

Apply the distributive property.

$y = - 3 (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) + 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = - 3 (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) + 4$

Move $- 5$ to the left of $x$ .

$y = - 3 (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) + 4$

Multiply $- 5$ by $- 5$ .

$y = - 3 (x^{2} - 5 x - 5 x + 25) + 4$

Subtract $5 x$ from $- 5 x$ .

$y = - 3 (x^{2} - 10 x + 25) + 4$

Apply the distributive property.

$y = - 3 x^{2} - 3 (- 10 x) - 3 \cdot 25 + 4$

Simplify.

Multiply $- 10$ by $- 3$ .

$y = - 3 x^{2} + 30 x - 3 \cdot 25 + 4$

Multiply $- 3$ by $25$ .

$y = - 3 x^{2} + 30 x - 75 + 4$

Add $- 75$ and $4$ .

$y = - 3 x^{2} + 30 x - 71$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - 3 x^{2} + 30 x - 71$

Vertex Form: $y = (- 3) {(x - (5))}^{2} + 4$

Step 6

Simplify the standard form.

Standard Form: $y = - 3 x^{2} + 30 x - 71$

Vertex Form: $y = - 3 {(x - 5)}^{2} + 4$

Step 7

Find the Parabola Through (6,1) with Vertex (5,4) (5,4) , (6,1)

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