Algebra Examples

$(2, 4)$ and $(4, 8)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(2, 4)$ as the vertex $(h, k)$ and $(4, 8)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$8 = a {(4 - (2))}^{2} + 4$

Step 2

Using $8 = a {(4 - (2))}^{2} + 4$ to solve for $a$ , $a = 1$ .

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Step 2.1

Rewrite the equation as $a {(4 - (2))}^{2} + 4 = 8$ .

$a {(4 - (2))}^{2} + 4 = 8$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $- 1$ by $2$ .

$a {(4 - 2)}^{2} + 4 = 8$

Step 2.2.2

Subtract $2$ from $4$ .

$a \cdot 2^{2} + 4 = 8$

Step 2.2.3

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

$a \cdot 4 + 4 = 8$

Step 2.2.4

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

$4 a + 4 = 8$

Step 2.3

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

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Step 2.3.1

Subtract $4$ from both sides of the equation.

$4 a = 8 - 4$

Step 2.3.2

Subtract $4$ from $8$ .

$4 a = 4$

Step 2.4

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

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Step 2.4.1

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

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

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.

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

Step 2.4.2.1.2

Divide $a$ by $1$ .

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $4$ by $4$ .

$a = 1$

Step 3

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

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

Step 4

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

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Step 4.1

Remove parentheses.

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

Step 4.2

Multiply $1$ by ${(x - (2))}^{2}$ .

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

Step 4.3

Remove parentheses.

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

Step 4.4

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

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Multiply ${(x - (2))}^{2}$ by $1$ .

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

Step 4.4.1.2

Multiply $- 1$ by $2$ .

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

Step 4.4.1.3

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

$y = (x - 2) (x - 2) + 4$

Step 4.4.1.4

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 4.4.1.4.1

Apply the distributive property.

$y = x (x - 2) - 2 (x - 2) + 4$

Step 4.4.1.4.2

Apply the distributive property.

$y = x \cdot x + x \cdot - 2 - 2 (x - 2) + 4$

Step 4.4.1.4.3

Apply the distributive property.

$y = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 + 4$

Step 4.4.1.5

Simplify and combine like terms.

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Step 4.4.1.5.1

Simplify each term.

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Step 4.4.1.5.1.1

Multiply $x$ by $x$ .

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

Step 4.4.1.5.1.2

Move $- 2$ to the left of $x$ .

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

Step 4.4.1.5.1.3

Multiply $- 2$ by $- 2$ .

$y = x^{2} - 2 x - 2 x + 4 + 4$

Step 4.4.1.5.2

Subtract $2 x$ from $- 2 x$ .

$y = x^{2} - 4 x + 4 + 4$

Step 4.4.2

Add $4$ and $4$ .

$y = x^{2} - 4 x + 8$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = x^{2} - 4 x + 8$

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

Step 6

Simplify the standard form.

Standard Form: $y = x^{2} - 4 x + 8$

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

Step 7