Precalculus Examples

$f (x) = 2 x - x^{2} + 8$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 2 x - x^{2} + 8$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x - x^{2} + 8 = 0$ .

$2 x - x^{2} + 8 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$2 u - u^{2} + 8 = 0$

Step 1.2.2.2

Factor by grouping.

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Step 1.2.2.2.1

Reorder terms.

$- u^{2} + 2 u + 8 = 0$

Step 1.2.2.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 8 = - 8$ and whose sum is $b = 2$ .

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Step 1.2.2.2.2.1

Factor $2$ out of $2 u$ .

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

Step 1.2.2.2.2.2

Rewrite $2$ as $- 2$ plus $4$

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

Step 1.2.2.2.2.3

Apply the distributive property.

$- u^{2} - 2 u + 4 u + 8 = 0$

Step 1.2.2.2.3

Factor out the greatest common factor from each group.

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Step 1.2.2.2.3.1

Group the first two terms and the last two terms.

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

Step 1.2.2.2.3.2

Factor out the greatest common factor (GCF) from each group.

$u (- u - 2) - 4 (- u - 2) = 0$

Step 1.2.2.2.4

Factor the polynomial by factoring out the greatest common factor, $- u - 2$ .

$(- u - 2) (u - 4) = 0$

Step 1.2.2.3

Replace all occurrences of $u$ with $x$ .

$(- x - 2) (x - 4) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$- x - 2 = 0$

$x - 4 = 0$

Step 1.2.4

Set $- x - 2$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $- x - 2$ equal to $0$ .

$- x - 2 = 0$

Step 1.2.4.2

Solve $- x - 2 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $2$ to both sides of the equation.

$- x = 2$

Step 1.2.4.2.2

Divide each term in $- x = 2$ by $- 1$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $- x = 2$ by $- 1$ .

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

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.4.2.2.2.2

Divide $x$ by $1$ .

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

Step 1.2.4.2.2.3

Simplify the right side.

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Step 1.2.4.2.2.3.1

Divide $2$ by $- 1$ .

$x = - 2$

Step 1.2.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 1.2.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.6

The final solution is all the values that make $(- x - 2) (x - 4) = 0$ true.

$x = - 2, 4$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0), (4, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Simplify $2 (0) - {(0)}^{2} + 8$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $2$ by $0$ .

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

Step 2.2.3.1.2

Raising $0$ to any positive power yields $0$ .

$y = 0 - 0 + 8$

Step 2.2.3.1.3

Multiply $- 1$ by $0$ .

$y = 0 + 0 + 8$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 8$

Step 2.2.3.2.2

Add $0$ and $8$ .

$y = 8$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 8)$

Step 3

List the intersections.

x-intercept(s): $(- 2, 0), (4, 0)$

y-intercept(s): $(0, 8)$

Step 4