Algebra Examples

$(0, 8)$ , $(4, 0)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(0, 8)$ as the vertex $(h, k)$ and $(4, 0)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$0 = a {(4 - (0))}^{2} + 8$

Step 2

Using $0 = a {(4 - (0))}^{2} + 8$ to solve for $a$ , $a = - \frac{1}{2}$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $a {(4 - (0))}^{2} + 8 = 0$ .

$a {(4 - (0))}^{2} + 8 = 0$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Subtract $0$ from $4$ .

$a \cdot 4^{2} + 8 = 0$

Step 2.2.2

Raise $4$ to the power of $2$ .

$a \cdot 16 + 8 = 0$

Step 2.2.3

Move $16$ to the left of $a$ .

$16 a + 8 = 0$

Step 2.3

Subtract $8$ from both sides of the equation.

$16 a = - 8$

Step 2.4

Divide each term in $16 a = - 8$ by $16$ and simplify.

Tap for more steps...

Step 2.4.1

Divide each term in $16 a = - 8$ by $16$ .

$\frac{16 a}{16} = \frac{- 8}{16}$

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $16$ .

Tap for more steps...

Step 2.4.2.1.1

Cancel the common factor.

$\frac{16 a}{16} = \frac{- 8}{16}$

Step 2.4.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 8}{16}$

Step 2.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.3.1

Cancel the common factor of $- 8$ and $16$ .

Tap for more steps...

Step 2.4.3.1.1

Factor $8$ out of $- 8$ .

$a = \frac{8 (- 1)}{16}$

Step 2.4.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.4.3.1.2.1

Factor $8$ out of $16$ .

$a = \frac{8 \cdot - 1}{8 \cdot 2}$

Step 2.4.3.1.2.2

Cancel the common factor.

$a = \frac{8 \cdot - 1}{8 \cdot 2}$

Step 2.4.3.1.2.3

Rewrite the expression.

$a = \frac{- 1}{2}$

Step 2.4.3.2

Move the negative in front of the fraction.

$a = - \frac{1}{2}$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(0, 8)$ and $a = - \frac{1}{2}$ is $y = (- \frac{1}{2}) {(x - (0))}^{2} + 8$ .

$y = (- \frac{1}{2}) {(x - (0))}^{2} + 8$

Step 4

Solve $y = (- \frac{1}{2}) {(x - (0))}^{2} + 8$ for $y$ .

Tap for more steps...

Step 4.1

Remove parentheses.

$y = (- \frac{1}{2}) {(x - (0))}^{2} + 8$

Step 4.2

Multiply $- \frac{1}{2}$ by ${(x - (0))}^{2}$ .

$y = - \frac{1}{2} \cdot {(x - (0))}^{2} + 8$

Step 4.3

Remove parentheses.

$y = (- \frac{1}{2}) {(x - (0))}^{2} + 8$

Step 4.4

Simplify each term.

Tap for more steps...

Step 4.4.1

Subtract $0$ from $x$ .

$y = - \frac{1}{2} x^{2} + 8$

Step 4.4.2

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

$y = - \frac{x^{2}}{2} + 8$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - \frac{1}{2} x^{2} + 8$

Vertex Form: $y = (- \frac{1}{2}) {(x - (0))}^{2} + 8$

Step 6

Simplify the standard form.

Standard Form: $y = - \frac{1}{2} x^{2} + 8$

Vertex Form: $y = - \frac{1}{2} x^{2} + 8$

Step 7