Precalculus Examples

$f (x) = \frac{x^{2} - 3 x - 4}{x^{2} + 8 x + 7}$

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 $x$ .

$0 = \frac{x^{2} - 3 x - 4}{x^{2} + 8 x + 7}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{2} - 3 x - 4 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor $x^{2} - 3 x - 4$ using the AC method.

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Step 1.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 1.2.2.1.2

Write the factored form using these integers.

$(x - 4) (x + 1) = 0$

Step 1.2.2.2

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

$x - 4 = 0$

$x + 1 = 0$

Step 1.2.2.3

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

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Step 1.2.2.3.1

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

$x - 4 = 0$

Step 1.2.2.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 1.2.2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.2.5

The final solution is all the values that make $(x - 4) (x + 1) = 0$ true.

$x = 4, - 1$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x^{2} - 3 x - 4}{x^{2} + 8 x + 7}$ true.

$x = 4$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = \frac{{(0)}^{2} - 3 \cdot 0 - 4}{{(0)}^{2} + 8 (0) + 7}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0^{2} - 3 \cdot 0 - 4}{{(0)}^{2} + 8 (0) + 7}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{2} - 3 \cdot 0 - 4}{0^{2} + 8 (0) + 7}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{2} - 3 \cdot 0 - 4}{{(0)}^{2} + 8 (0) + 7}$

Step 2.2.4

Simplify $\frac{{(0)}^{2} - 3 \cdot 0 - 4}{{(0)}^{2} + 8 (0) + 7}$ .

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

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

$y = \frac{0 - 3 \cdot 0 - 4}{0^{2} + 8 (0) + 7}$

Step 2.2.4.1.2

Multiply $- 3$ by $0$ .

$y = \frac{0 + 0 - 4}{0^{2} + 8 (0) + 7}$

Step 2.2.4.1.3

Add $0$ and $0$ .

$y = \frac{0 - 4}{0^{2} + 8 (0) + 7}$

Step 2.2.4.1.4

Subtract $4$ from $0$ .

$y = \frac{- 4}{0^{2} + 8 (0) + 7}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

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

$y = \frac{- 4}{0 + 8 (0) + 7}$

Step 2.2.4.2.2

Multiply $8$ by $0$ .

$y = \frac{- 4}{0 + 0 + 7}$

Step 2.2.4.2.3

Add $0$ and $0$ .

$y = \frac{- 4}{0 + 7}$

Step 2.2.4.2.4

Add $0$ and $7$ .

$y = \frac{- 4}{7}$

Step 2.2.4.3

Move the negative in front of the fraction.

$y = - \frac{4}{7}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{4}{7})$

Step 3

List the intersections.

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

y-intercept(s): $(0, - \frac{4}{7})$

Step 4