Math / Algebra

QuestionFollow the steps for graphing a rational function to graph the function $R(x)=\frac{x^{2}+10 x+16}{x+8}$ . (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) C. The graph has neither $x$ -intercepts nor $y$ -intercepts. D. The graph has $x$ -intercept(s) $\square$ and no $y$ -intercept(s). (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) Determine the behavior of the graph of $R$ at any $x$ -intercepts. Select the correct choice and, if necessary, fill in the answer box(es) within your choice. A. The graph will cross the $x$ -axis at $x=$ $\square$ and touch the $x$ -axis at $x=$ $\square$ (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) B. The graph will cross the $x$ -axis at $x=-2$ . $\square$ (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) C. The graph will touch the $x$ -axis at $x=$ $\square$ (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) D. There is no x-intercept.
Determine the vertical asymptote(s), if any exist. Select the correct choice and, if necessary, fill in the answer box(es) within your choice. A. The function has one vertical asymptote, $\square$ . (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two vertical asymptotes. The leftmost asymptote is $\square$ , and the rightmost asymptote is $\square$ (Type equations. Use integers or fractions for any numbers in the equations.) C. The function has three vertical asymptotes. The leftmost asymptote is $\square$ , the middle asymptote is $\square$ , and the rightmost asymptote is $\square$ . (Type equations. Use integers or fractions for any numbers in the equations.) D. The function has no vertical asymptote.
Determine the hole, if it exists. Select the correct choice and, if necessary, fill in the answer box to complete your choice.
There is a hole in the graph at the point $\square$ $\square$ . (Type an ordered pair using integers or fractions.) B. There are no holes in the graph.

Studdy Solution

STEP 1

What is this asking? We need to graph the rational function $R(x) = \frac{x^2 + 10x + 16}{x + 8}$ , find its x-intercepts, y-intercepts, vertical asymptotes, and holes. Watch out! Remember that a hole occurs when a factor cancels out in the numerator and denominator, while a vertical asymptote occurs when the denominator is zero and the numerator is not zero at the same $x$ value.

STEP 2

1. Simplify the function
2. Find x-intercepts
3. Find y-intercepts
4. Find vertical asymptotes
5. Find holes

STEP 3

Let's factor the numerator of our function $R(x)$ .
We're looking for two numbers that multiply to 16 and add up to 10.
Those numbers are 2 and 8!
So, the numerator becomes $(x+2)(x+8)$ .

STEP 4

Now, our function looks like this: $R(x) = \frac{(x+2)(x+8)}{x+8}$ .
Since we have the factor $(x+8)$ in both the numerator and the denominator, and $x$ cannot be equal to $-8$ , we can divide both numerator and denominator by $(x+8)$ , effectively multiplying by one.
This simplifies our function to $R(x) = x+2$ for $x \ne -8$ .

STEP 5

To find the x-intercept, we set $R(x) = 0$ , so $x+2 = 0$ .

STEP 6

Subtracting 2 from both sides gives us $x = -2$ .
This means the graph will cross the x-axis at $x = -2$ .

STEP 7

To find the y-intercept, we set $x = 0$ in our simplified function: $R(0) = 0 + 2 = 2$ .

STEP 8

Since the simplified function $R(x) = x+2$ has no denominator, there are no vertical asymptotes.

STEP 9

Remember that we canceled out the factor $(x+8)$ earlier.
This means there's a hole at $x = -8$ .

STEP 10

To find the y-coordinate of the hole, we plug $x = -8$ into the simplified function: $R(-8) = -8 + 2 = -6$ .

STEP 11

So, the hole is at the point $(-8, -6)$ .

STEP 12

D. The graph has x-intercept(s) $-2$ and no y-intercept(s). B. The graph will cross the x-axis at $x=-2$ . D. The function has no vertical asymptote. There is a hole in the graph at the point $(-8, -6)$ .

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Studdy Solution

STEP 1

STEP 2

1. Simplify the function
2. Find x-intercepts
3. Find y-intercepts
4. Find vertical asymptotes
5. Find holes

STEP 3

Let's factor the numerator of our function $R(x)$ .
We're looking for two numbers that multiply to 16 and add up to 10.
Those numbers are 2 and 8!
So, the numerator becomes $(x+2)(x+8)$ .

STEP 4

STEP 5

To find the x-intercept, we set $R(x) = 0$ , so $x+2 = 0$ .

STEP 6

Subtracting 2 from both sides gives us $x = -2$ .
This means the graph will cross the x-axis at $x = -2$ .

STEP 7

To find the y-intercept, we set $x = 0$ in our simplified function: $R(0) = 0 + 2 = 2$ .

STEP 8

Since the simplified function $R(x) = x+2$ has no denominator, there are no vertical asymptotes.

STEP 9

Remember that we canceled out the factor $(x+8)$ earlier.
This means there's a hole at $x = -8$ .

STEP 10

To find the y-coordinate of the hole, we plug $x = -8$ into the simplified function: $R(-8) = -8 + 2 = -6$ .

STEP 11

So, the hole is at the point $(-8, -6)$ .

STEP 12

D. The graph has x-intercept(s) $-2$ and no y-intercept(s). B. The graph will cross the x-axis at $x=-2$ . D. The function has no vertical asymptote. There is a hole in the graph at the point $(-8, -6)$ .

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Studdy Solution

STEP 1

STEP 2

1. Simplify the function2. Find x-intercepts3. Find y-intercepts4. Find vertical asymptotes5. Find holes

STEP 3

Let's **factor** the numerator of our function R(x)R(x)R(x).We're looking for two numbers that multiply to **16** and add up to **10**.Those numbers are **2** and **8**!So, the numerator becomes (x+2)(x+8)(x+2)(x+8)(x+2)(x+8).

STEP 4

STEP 5

To find the x-intercept, we set R(x)=0R(x) = 0R(x)=0, so x+2=0x+2 = 0x+2=0.

STEP 6

Subtracting **2** from both sides gives us x=−2x = -2x=−2.This means the graph will cross the x-axis at x=−2x = -2x=−2.

STEP 7

To find the y-intercept, we set x=0x = 0x=0 in our simplified function: R(0)=0+2=2R(0) = 0 + 2 = 2R(0)=0+2=2.

STEP 8

Since the simplified function R(x)=x+2R(x) = x+2R(x)=x+2 has no denominator, there are no vertical asymptotes.

STEP 9

Remember that we canceled out the factor (x+8)(x+8)(x+8) earlier.This means there's a hole at x=−8x = -8x=−8.

STEP 10

To find the y-coordinate of the hole, we plug x=−8x = -8x=−8 into the simplified function: R(−8)=−8+2=−6R(-8) = -8 + 2 = -6R(−8)=−8+2=−6.

STEP 11

So, the hole is at the point (−8,−6)(-8, -6)(−8,−6).

STEP 12

D. The graph has x-intercept(s) −2-2−2 and no y-intercept(s). B. The graph will cross the x-axis at x=−2x=-2x=−2. D. The function has no vertical asymptote. There is a hole in the graph at the point (−8,−6)(-8, -6)(−8,−6).

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1. Simplify the function
2. Find x-intercepts
3. Find y-intercepts
4. Find vertical asymptotes
5. Find holes

Let's factor the numerator of our function $R(x)$ .
We're looking for two numbers that multiply to 16 and add up to 10.
Those numbers are 2 and 8!
So, the numerator becomes $(x+2)(x+8)$ .

To find the x-intercept, we set $R(x) = 0$ , so $x+2 = 0$ .

Subtracting 2 from both sides gives us $x = -2$ .
This means the graph will cross the x-axis at $x = -2$ .

To find the y-intercept, we set $x = 0$ in our simplified function: $R(0) = 0 + 2 = 2$ .

Since the simplified function $R(x) = x+2$ has no denominator, there are no vertical asymptotes.

Remember that we canceled out the factor $(x+8)$ earlier.
This means there's a hole at $x = -8$ .

To find the y-coordinate of the hole, we plug $x = -8$ into the simplified function: $R(-8) = -8 + 2 = -6$ .

So, the hole is at the point $(-8, -6)$ .

D. The graph has x-intercept(s) $-2$ and no y-intercept(s). B. The graph will cross the x-axis at $x=-2$ . D. The function has no vertical asymptote. There is a hole in the graph at the point $(-8, -6)$ .