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Question Find the x-intercept, y-intercept, and rotational symmetry of the function f(x)=−(x+3)3+2f(x)=-(x+3)^{3}+2.

Studdy Solution

STEP 1

Assumptions
1. The function given is f(x)=−(x+3)3+2 f(x) = -(x+3)^3 + 2 .
2. To find the x-intercept, we need to set f(x)=0 f(x) = 0 and solve for x x .
3. To find the y-intercept, we need to evaluate f(0) f(0) .
4. To find the rotational symmetry, we need to determine if there's a point through which the function can be rotated 180 degrees and remain unchanged.

STEP 2

Find the x-intercept by setting f(x)=0 f(x) = 0 .
0=−(x+3)3+2 0 = -(x+3)^3 + 2

STEP 3

Move the constant term to the other side of the equation.
(x+3)3=2 (x+3)^3 = 2

STEP 4

Take the cube root of both sides to solve for x x .
x+3=23 x+3 = \sqrt[3]{2}

STEP 5

Isolate x x to find the x-intercept.
x=23−3 x = \sqrt[3]{2} - 3
The x-intercept is x=23−3 x = \sqrt[3]{2} - 3 .

STEP 6

Find the y-intercept by evaluating f(0) f(0) .
f(0)=−(0+3)3+2 f(0) = - (0+3)^3 + 2

STEP 7

Simplify the expression to find the y-intercept.
f(0)=−(3)3+2 f(0) = - (3)^3 + 2
f(0)=−27+2 f(0) = - 27 + 2
f(0)=−25 f(0) = - 25
The y-intercept is y=−25 y = -25 .

STEP 8

To determine rotational symmetry, we need to check if the function is unchanged when x x is replaced with −x -x and vice versa. This is typically the case for odd functions, where f(−x)=−f(x) f(-x) = -f(x) , or even functions, where f(−x)=f(x) f(-x) = f(x) .

STEP 9

Replace x x with −x -x in the function.
f(−x)=−(−x+3)3+2 f(-x) = -(-x+3)^3 + 2

STEP 10

Simplify the expression.
f(−x)=−(x−3)3+2 f(-x) = -(x-3)^3 + 2

STEP 11

Compare f(−x) f(-x) with f(x) f(x) .
The function f(x)=−(x+3)3+2 f(x) = -(x+3)^3 + 2 is not equal to f(−x)=−(x−3)3+2 f(-x) = -(x-3)^3 + 2 . Therefore, the function does not have symmetry about the y-axis or the origin.

STEP 12

Since the function is a cubic function with a negative leading coefficient, it has rotational symmetry about the point where the function changes direction, which is the inflection point.

STEP 13

To find the inflection point, we need to find the second derivative and set it equal to zero.
First, find the first derivative of f(x) f(x) .
f′(x)=−3(x+3)2 f'(x) = -3(x+3)^2

STEP 14

Now, find the second derivative of f(x) f(x) .
f′′(x)=−6(x+3) f''(x) = -6(x+3)

STEP 15

Set the second derivative equal to zero to find the inflection point.
−6(x+3)=0 -6(x+3) = 0

STEP 16

Solve for x x to find the x-coordinate of the inflection point.
x=−3 x = -3

STEP 17

Find the y-coordinate of the inflection point by evaluating f(−3) f(-3) .
f(−3)=−(−3+3)3+2 f(-3) = -(-3+3)^3 + 2
f(−3)=−0+2 f(-3) = -0 + 2
f(−3)=2 f(-3) = 2

STEP 18

The inflection point is at (−3,2) (-3, 2) .
The function has rotational symmetry about the inflection point (−3,2) (-3, 2) .
x-intercept =23−3 = \sqrt[3]{2} - 3 y-intercept =−25 = -25 rotational symmetry about (−3,2) (-3, 2)

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