Math / Algebra

Question.4-4.6)
Graph the function $y=\frac{x^{2}-63}{x-8}$ by identifying the domain and any where the curve is increasing and where it is decreasing, finding as intercepts, critical points, and inflection points. Then find coord

Studdy Solution

STEP 1

1. The function given is $y = \frac{x^2 - 63}{x - 8}$ .
2. We need to graph the function by identifying the domain, intervals of increase and decrease, intercepts, critical points, and inflection points.

STEP 2

1. Determine the domain of the function.
2. Find the intercepts of the function.
3. Identify critical points and intervals of increase and decrease.
4. Determine inflection points and concavity.
5. Graph the function using the information gathered.

STEP 3

To find the domain, identify values of $x$ that make the denominator zero. The denominator is $x - 8$ .
Set $x - 8 = 0$ to find the restriction: $x = 8$
Thus, the domain is all real numbers except $x = 8$ .

STEP 4

To find the y-intercept, set $x = 0$ in the function: $y = \frac{0^2 - 63}{0 - 8} = \frac{-63}{-8} = \frac{63}{8}$
The y-intercept is $\left(0, \frac{63}{8}\right)$ .
To find the x-intercepts, set $y = 0$ : $\frac{x^2 - 63}{x - 8} = 0$
This implies $x^2 - 63 = 0$ , so: $x^2 = 63$ $x = \pm \sqrt{63}$
The x-intercepts are $(\sqrt{63}, 0)$ and $(-\sqrt{63}, 0)$ .

STEP 5

To find critical points, take the derivative of $y$ using the quotient rule: $y' = \frac{(2x)(x-8) - (x^2 - 63)(1)}{(x-8)^2}$
Simplify the numerator: $2x^2 - 16x - x^2 + 63 = x^2 - 16x + 63$
Set the derivative equal to zero to find critical points: $x^2 - 16x + 63 = 0$
Solve the quadratic equation: $(x - 9)(x - 7) = 0$
Critical points are $x = 9$ and $x = 7$ .
Determine intervals of increase and decrease by testing intervals around critical points.

STEP 6

Test intervals: - For $x < 7$ , choose $x = 6$ . - For $7 < x < 9$ , choose $x = 8$ . - For $x > 9$ , choose $x = 10$ .
Evaluate the sign of $y'$ in each interval to determine increasing/decreasing behavior.

STEP 7

To find inflection points, take the second derivative $y''$ and set it to zero.
Calculate $y''$ and solve for $x$ to find inflection points.
Determine concavity by testing intervals around inflection points.

STEP 8

Use the domain, intercepts, critical points, intervals of increase/decrease, and inflection points to sketch the graph of the function.
The function $y = \frac{x^2 - 63}{x - 8}$ has been analyzed and can now be graphed based on the information obtained.

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Question.4-4.6)
Graph the function $y=\frac{x^{2}-63}{x-8}$ by identifying the domain and any where the curve is increasing and where it is decreasing, finding as intercepts, critical points, and inflection points. Then find coord

Studdy Solution

STEP 1

1. The function given is $y = \frac{x^2 - 63}{x - 8}$ .
2. We need to graph the function by identifying the domain, intervals of increase and decrease, intercepts, critical points, and inflection points.

STEP 2

1. Determine the domain of the function.
2. Find the intercepts of the function.
3. Identify critical points and intervals of increase and decrease.
4. Determine inflection points and concavity.
5. Graph the function using the information gathered.

STEP 3

To find the domain, identify values of $x$ that make the denominator zero. The denominator is $x - 8$ .
Set $x - 8 = 0$ to find the restriction: $x = 8$
Thus, the domain is all real numbers except $x = 8$ .

STEP 4

STEP 5

STEP 6

Test intervals: - For $x < 7$ , choose $x = 6$ . - For $7 < x < 9$ , choose $x = 8$ . - For $x > 9$ , choose $x = 10$ .
Evaluate the sign of $y'$ in each interval to determine increasing/decreasing behavior.

STEP 7

To find inflection points, take the second derivative $y''$ and set it to zero.
Calculate $y''$ and solve for $x$ to find inflection points.
Determine concavity by testing intervals around inflection points.

STEP 8

Use the domain, intercepts, critical points, intervals of increase/decrease, and inflection points to sketch the graph of the function.
The function $y = \frac{x^2 - 63}{x - 8}$ has been analyzed and can now be graphed based on the information obtained.

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Question.4-4.6)Graph the function y=x2−63x−8y=\frac{x^{2}-63}{x-8}y=x−8x2−63​ by identifying the domain and any where the curve is increasing and where it is decreasing, finding as intercepts, critical points, and inflection points. Then find coord

Studdy Solution

STEP 1

1. The function given is y=x2−63x−8 y = \frac{x^2 - 63}{x - 8} y=x−8x2−63​.2. We need to graph the function by identifying the domain, intervals of increase and decrease, intercepts, critical points, and inflection points.

STEP 2

1. Determine the domain of the function.2. Find the intercepts of the function.3. Identify critical points and intervals of increase and decrease.4. Determine inflection points and concavity.5. Graph the function using the information gathered.

STEP 3

To find the domain, identify values of x x x that make the denominator zero. The denominator is x−8 x - 8 x−8.Set x−8=0 x - 8 = 0 x−8=0 to find the restriction: x=8 x = 8 x=8Thus, the domain is all real numbers except x=8 x = 8 x=8.

STEP 4

STEP 5

STEP 6

Test intervals: - For x<7 x < 7 x<7, choose x=6 x = 6 x=6. - For 7<x<9 7 < x < 9 7<x<9, choose x=8 x = 8 x=8. - For x>9 x > 9 x>9, choose x=10 x = 10 x=10.Evaluate the sign of y′ y' y′ in each interval to determine increasing/decreasing behavior.

STEP 7

To find inflection points, take the second derivative y′′ y'' y′′ and set it to zero.Calculate y′′ y'' y′′ and solve for x x x to find inflection points.Determine concavity by testing intervals around inflection points.

STEP 8

Use the domain, intercepts, critical points, intervals of increase/decrease, and inflection points to sketch the graph of the function. The function y=x2−63x−8 y = \frac{x^2 - 63}{x - 8} y=x−8x2−63​ has been analyzed and can now be graphed based on the information obtained.

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Studdy solves anything!

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Question.4-4.6)
Graph the function $y=\frac{x^{2}-63}{x-8}$ by identifying the domain and any where the curve is increasing and where it is decreasing, finding as intercepts, critical points, and inflection points. Then find coord

1. The function given is $y = \frac{x^2 - 63}{x - 8}$ .
2. We need to graph the function by identifying the domain, intervals of increase and decrease, intercepts, critical points, and inflection points.

1. Determine the domain of the function.
2. Find the intercepts of the function.
3. Identify critical points and intervals of increase and decrease.
4. Determine inflection points and concavity.
5. Graph the function using the information gathered.

To find the domain, identify values of $x$ that make the denominator zero. The denominator is $x - 8$ .
Set $x - 8 = 0$ to find the restriction: $x = 8$
Thus, the domain is all real numbers except $x = 8$ .

Test intervals: - For $x < 7$ , choose $x = 6$ . - For $7 < x < 9$ , choose $x = 8$ . - For $x > 9$ , choose $x = 10$ .
Evaluate the sign of $y'$ in each interval to determine increasing/decreasing behavior.

To find inflection points, take the second derivative $y''$ and set it to zero.
Calculate $y''$ and solve for $x$ to find inflection points.
Determine concavity by testing intervals around inflection points.

Use the domain, intercepts, critical points, intervals of increase/decrease, and inflection points to sketch the graph of the function.
The function $y = \frac{x^2 - 63}{x - 8}$ has been analyzed and can now be graphed based on the information obtained.